📗 Class 9 (Maharashtra Board) — Chapter 3: Polynomials
Mobile-friendly • Comic Sans MS • MathJax
Copy-paste ready. Split into parts to keep your website menu bar unchanged.
Part A • 1-Mark Questions (20) — Red Q / Green A
1) Define a polynomial in one variable.
An algebraic expression \(a_n x^n+\cdots+a_1x+a_0\) with \(n\in\mathbb{W}\) and all exponents whole numbers; \(a_n\neq0\).
2) Is \(y+\dfrac{1}{y}\) a polynomial?
No. Term \(\dfrac{1}{y}=y^{-1}\) has negative exponent.
3) Give an example of binomial of degree \(5\).
\(x^5+1\) (or \(2x^5-3\)).
4) State degree of \(2x^7-5x+9\).
\(7\).
5) Degree of non-zero constant polynomial.
\(0\).
6) Degree of zero polynomial.
Not defined.
7) Write coefficient form of \(x^3+3x-5\).
\((1,0,3,-5)\).
8) Standard form of \(p(x)=x-3x^2+5+x^4\).
\(x^4-3x^2+x+5\).
9) Index form of \((4,0,-5,0,1)\) in variable \(y\).
\(4y^4+0y^3-5y^2+0y+1\).
10) Classify \(3x^2+5x+1\).
Quadratic trinomial.
11) Highest power in \(3m^3n^6+7m^2n^3-mn\) (degree).
\(9\) (from \(m^3n^6\)).
12) Value \(p(2)\) for \(p(x)=2x^2-3x+5\).
\(7\).
13) Remainder when \(p(x)\) is divided by \(x-a\) (name theorem).
Remainder \(=p(a)\) (Remainder Theorem).
14) Factor theorem (statement).
\((x-a)\) is a factor of \(p(x)\iff p(a)=0\).
15) Perform: \(-2a\cdot 5a^2\).
\(-10a^3\).
16) Add: \((3m^2n+5mn^2-7mn)+(2m^2n-mn^2+mn)\).
\(5m^2n+4mn^2-6mn\).
17) Divide: \((2x^2+2)\div(x+2)\) — state remainder.
Remainder \(=10\).
18) Synthetic division is applicable when divisor is of which form?
Linear: \(x+a\) or \(x-a\).
19) Degree of \(xyz+xy-z\).
\(3\).
20) Write three monomials of degree \(5\).
\(x^5,\ 2y^5,\ -7m^5\).
Part B • 2-Mark Questions (20) — Red Q / Green A
1) State whether each is a polynomial. Justify: (i) \(y+\dfrac1y\) (ii) \(2-\dfrac{5}{x}\) (iii) \(x^2+7x+9\) (iv) \(2m^{-2}+7m-5\) (v) \(10\).
(i) No (\(y^{-1}\)). (ii) No (\(x^{-1}\)). (iii) Yes. (iv) No (negative power). (v) Yes (constant).
2) Coefficient of \(m^3\): (i) \(m^3\) (ii) \(-\dfrac{3}{2}+m-3m^3\) (iii) \(-\dfrac{2}{3}m^3-5m^2+7m-1\).
(i) \(1\). (ii) \(-3\). (iii) \(-\dfrac{2}{3}\).
3) Write polynomials in \(x\): (i) monomial degree \(7\) (ii) binomial degree \(35\) (iii) trinomial degree \(8\).
(i) \(x^7\). (ii) \(x^{35}+1\). (iii) \(x^8+x+1\).
4) Degrees: (i) \(5\) (ii) \(x^0\) (iii) \(x^2\) (iv) \(2m^{10}-7\) (vii) \(xyz+xy-z\) (vi) \(7y-y^3+y^5\) (viii) \(m^3n^7-3m^5n+mn\).
(i) 0 (ii) 0 (iii) 2 (iv) 10 (vii) 3 (vi) 5 (viii) 10.
5) Classify: (i) \(2x^2+3x+1\) (ii) \(5p\) (iii) \(\dfrac{2}{m}-7\) (iv) \(m^3+7m^2+5\) (v) \(a^2\).
(i) Quadratic (ii) Linear (iii) Not a polynomial (iv) Cubic (v) Quadratic.
6) Standard forms: (i) \(m^3+3+5m\) (ii) \(-7y+y^5+3y^3-\dfrac12+2y^4-y^2\).
(i) \(m^3+5m+3\). (ii) \(y^5+2y^4+3y^3-y^2-7y-\dfrac12\).
7) Coefficient forms: (i) \(x^3-2\) (ii) \(5y\).
(i) \((1,0,0,-2)\). (ii) \((5,0)\).
8) Index forms in \(x\): (i) \((1,2,3)\) (ii) \((5,0,0,0,-1)\) (iii) \((-2,2,-2,2)\).
(i) \(x^2+2x+3\). (ii) \(5x^4-1\). (iii) \(-2x^3+2x^2-2x+2\).
9) Add: \((-7m^4+5m^3+2)\) and \((5m^4-3m^3+2m^2+3m-6)\).
\((-2m^4+2m^3+2m^2+3m-4)\).
10) Subtract \((-19x+\tfrac{3}{x}? +7x^2)\) from \((x^2-9x+\tfrac{3}{?})\) (like-term subtraction idea).
Arrange like terms and subtract coefficients; (demonstration item — avoid non-polynomial parts on exams).
11) Multiply: \((m^2-5)(m^3+2m-2)\).
\(m^5-3m^3-2m^2-10m+10\).
12) Multiply: \(2x\) with \(x^2-2x-1\).
\(2x^3-4x^2-2x\).
13) Degree of product of degree \(3\) and degree \(5\) polynomials.
\(3+5=8\).
14) Divide: \((2x^2+2)\div(x+2)\). Give quotient and remainder.
Quotient \(=2x-4\), Remainder \(=10\). \(\ (2x^2+2)=(x+2)(2x-4)+10\).
15) Synthetic division: \((3x^3+2x^2-1)\div(x+2)\).
Quotient \(=3x^2-4x+8\), Remainder \(=-17\).
16) Find \(p(0)\) if \(p(x)=2x^2-x^3+x+2\).
\(p(0)=2\).
17) If \(p(m)=m^2-am+7\) and \(p(-1)=10\), find \(a\).
\(a=2\).
18) Remainder when \(x^4-5x^2-4x\) is divided by \(x+3\).
\(p(-3)=48\).
19) If \(x-1\) is a factor of \(x^3-2x^2+mx-4\), find \(m\).
\(m=5\).
20) Factorize \(4x^2-25\).
\((2x+5)(2x-5)\).
Part C • 3-Mark Questions (20) — Red Q / Green A
1) Add: \(x^3-2x^2-9\) and \(5x^3+2x+9\). Subtract the second from the first.
Sum \(=6x^3-2x^2+2x\). Difference \(=(x^3-2x^2-9)-(5x^3+2x+9)=-4x^3-2x^2-2x-18\).
2) Multiply: \((2y+1)(y^2-2y^3+3y)\).
\(-4y^4+y^2+6y^2+3y = -4y^4+7y^2+3y\).
3) Divide \((x^3-64)\) by \((x-4)\). Write \( \text{Dividend}=\text{Divisor}\cdot\text{Quotient}+\text{Remainder}\).
\((x^3-64)=(x-4)(x^2+4x+16)+0\).
4) Divide \(5x^5+4x^4-3x^3+2x^2+2\) by \(x^2-x\).
Quotient \(=5x^3+9x^2+6x+8\), Remainder \(=8x+2\).
5) Synthetic-then-linear check: \((2m^2-3m+10)\div(m-5)\).
Quotient \(=2m+7\), Remainder \(=45\).
6) Synthetic division: \((x^4+2x^3+3x^2+4x+5)\div(x+2)\).
Quotient \(=x^3+0x^2+3x-2\), Remainder \(=9\).
7) Synthetic division: \((y^3-216)\div(y-6)\).
Quotient \(=y^2+6y+36\), Remainder \(=0\).
8) Value: If \(p(y)=2y^3-6y^2-5y+7\), find \(p(2)\).
\(16-24-10+7=-11\).
9) Show that \(x-2\) is a factor of \(x^3-x^2-4\).
\(p(2)=8-4-4=0\Rightarrow\) factor by Factor Theorem.
10) Find \(k\) if \(t^3-3t^2+kt+50\) gives remainder \(62\) on division by \((t-3)\).
\(p(3)=27-27+3k+50=62\Rightarrow k=4\).
11) Factorize \((y^2-3y)^2-5(y^2-3y)-50\).
Let \(x=y^2-3y\). Then \(x^2-5x-50=(x-10)(x+5)=(y^2-3y-10)(y^2-3y+5)\) \(=(y-5)(y+2)(y^2-3y+5)\).
12) Factorize \((x+2)(x-3)(x-7)(x-2)+64\).
Let \(m=x^2-5x\). Then \((m-14)(m+6)+64=m^2-8m-20=(m-10)(m+2)\) \(=(x^2-5x-10)(x^2-5x+2)\).
13) Factorize \((x^2-x)^2-8(x^2-x)+12\).
\((x^2-x-2)(x^2-x-6)=(x-2)(x+1)(x-3)(x+2)\).
14) Factorize \((x-5)^2-(5x-25)-24\).
Let \(u=x-5\). Then \(u^2-5u-24=(u-8)(u+3)=(x-13)(x-2)\).
15) Factorize \((x^2-6x)^2-8(x^2-6x+8)-64\).
Let \(v=x^2-6x\). Then \(v^2-8v-128=(v-16)(v+8)=(x-8)(x+2)(x-2)(x-4)\).
16) Factorize \((x^2-2x+3)(x^2-2x+5)-35\).
Let \(w=x^2-2x\). Product \(=(w+3)(w+5)-35=w^2+8w-20=(w+10)(w-2)\) \(=(x^2-2x+10)(x^2-2x-2)\).
17) Factorize \((y+2)(y-3)(y+8)(y+3)+56\).
Let \(s=y^2+5y\). Then expression \(=s^2-18s-88=(s-22)(s+4)\) \(=(y^2+5y-22)(y^2+5y+4)=(y^2+5y-22)(y+1)(y+4)\).
18) Factorize \((y^2+5y)(y^2+5y-2)-24\).
Let \(s=y^2+5y\). Then \(s^2-2s-24=(s-6)(s+4)=(y+6)(y-1)(y+1)(y+4)\).
19) Simplify and state degree: \((x^2-3)(2x-7x^3+4)\).
\(-7x^5+2x^3+4x^2-6x+12\). Degree \(=5\).
20) If \(x+3\) divides \(x^3-2mx+21\), find \(m\).
Remainder \(p(-3)=0\Rightarrow -27+6m+21=0\Rightarrow m=1\).
Part D • Textbook Exercises — Perfect Solutions (Practice Sets 3.1 to 3.6 & Problem Set 3)
Practice Set 3.1 — Solutions
1) State whether each is a polynomial. \( \big(\)i\()\ y+\dfrac1y\) (ii) \(2-\dfrac{5}{x}\) (iii) \(x^2+7x+9\) (iv) \(2m^{-2}+7m-5\) (v) \(10\).
No, No, Yes, No, Yes.
2) Coefficient of \(m^3\) in: (i) \(m^3\) (ii) \(-\dfrac{3}{2}+m-3m^3\) (iii) \(-\dfrac{2}{3}m^3-5m^2+7m-1\).
\(1,\ -3,\ -\dfrac{2}{3}\).
3) Write polynomials in \(x\): (i) monomial, degree \(7\) (ii) binomial, degree \(35\) (iii) trinomial, degree \(8\).
\(x^7;\ x^{35}+1;\ x^8+x+1\).
4) Degrees of: (i) \(5\) (ii) \(x^0\) (iii) \(x^2\) (iv) \(2m^{10}-7\) (vi) \(7y-y^3+y^5\) (vii) \(xyz+xy-z\) (viii) \(m^3n^7-3m^5n+mn\).
\(0,0,2,10,5,3,10\).
5) Classify: (i) \(2x^2+3x+1\) (ii) \(5p\) (iii) \(\dfrac{2}{m}-7\) (iv) \(m^3+7m^2+5\) (v) \(a^2\).
Quadratic, Linear, Not a polynomial, Cubic, Quadratic.
6) Standard form: (i) \(m^3+3+5m\) (ii) \(-7y+y^5+3y^3-\dfrac12+2y^4-y^2\) (v) \(2p-\dfrac{2}{y}-\dfrac12\) (vi) \(3r^3\).
\(m^3+5m+3;\ y^5+2y^4+3y^3-y^2-7y-\dfrac12;\) already non-polynomial (keep as written); \(3r^3\).
7) Coefficient form: (i) \(x^3-2\) (ii) \(5y\).
\((1,0,0,-2);\ (5,0)\).
8) Index form: (i) \((1,2,3)\) (ii) \((5,0,0,0,-1)\) (iii) \(2m^4-3m^2+7\) (iv) \((-2,2,-2,2)\).
\(x^2+2x+3;\ 5x^4-1;\ 2m^4-3m^2+7;\ -2x^3+2x^2-2x+2.\)
9) Fill boxes (example answers):
Linear: \(x+7\); Quadratic: \(2x^2+5x+10\); Cubic: \(x^3+x^2+x+5\); Monomial: \(2x\); Binomial: \(x^3+9\); Trinomial: \(3x^2+5x\).
Practice Set 3.2 — Solutions
1) Model with letters: (i) a trees, +b per year, after x years?
\(\ a+bx\).
1(ii) \(y\) students per row, \(x\) rows ⇒ total?
\(\ xy\).
1(iii) Tens \(=m\), units \(=n\) ⇒ number?
\(\ 10m+n\).
2) Add: (i) \(x^3-2x^2-9\) and \(5x^3+2x+9\).
\(6x^3-2x^2+2x\).
2(ii) \(-7m^4+5m^3+2\) and \(5m^4-3m^3+2m^2+3m-6\).
\(-2m^4+2m^3+2m^2+3m-4\).
2(iii) \(2y^2+7y+5,\ 3y+9,\ 3y^2-4y-3\).
Sum \(=5y^2+6y+11\).
3) Subtract second from first: (i) \(x^2-9x+\tfrac{3}{\;}\) and \(-19x+\tfrac{3}{\;}\ +7x^2\) (illustrative).
Arrange like terms: \((x^2-7x^2)+(-9x+19x)+(\cdots)= -6x^2+10x+(\cdots)\).
3(ii) \(2ab^2+3a^2b-4ab\) minus \((3ab-8ab^2+2a^2b)\).
\((2ab^2+3a^2b-4ab)-(3ab-8ab^2+2a^2b)=10ab^2+a^2b-7ab\).
4) Multiply: (i) \(2x(x^2-2x-1)\) (ii) \((x^5-1)(x^3+2x^2+2)\) (iii) \((2y+1)(y^2-2y^3+3y)\).
(i) \(2x^3-4x^2-2x\). (ii) \(x^8+2x^7+2x^5-x^3-2x^2-2\). (iii) \(-4y^4+7y^2+3y\).
5) Divide and write \( \text{Dividend}=\text{Divisor}\cdot\text{Quotient}+\text{Remainder}\): (i) \((x^3-64)\div(x-4)\)
\((x-4)(x^2+4x+16)+0\).
5(ii) \((5x^5+4x^4-3x^3+2x^2+2)\div(x^2-x)\)
\((x^2-x)(5x^3+9x^2+6x+8)+(8x+2)\).
6*) Farm area: rectangle \(L=(2a^2+3b^2)\), \(B=(a^2+b^2)\); house is square of side \((a^2-b^2)\). Remaining area?
\((2a^2+3b^2)(a^2+b^2)-(a^2-b^2)^2= a^4+5a^2b^2+2b^4\).
Practice Set 3.3 — Synthetic & Linear Division
1) \((2m^2-3m+10)\div(m-5)\).
Quotient \(=2m+7\), Remainder \(=45\).
2) \((x^4+2x^3+3x^2+4x+5)\div(x+2)\).
Quotient \(=x^3+0x^2+3x-2\), Remainder \(=9\).
3) \((y^3-216)\div(y-6)\).
Quotient \(=y^2+6y+36\), Remainder \(=0\).
4) \((2x^4+3x^3-2x^2+4x)\div(x+3)\).
Quotient \(=2x^3-3x^2+7x-17\), Remainder \(=51\).
5) \((x^4-3x^2-8)\div(x+4)\).
Quotient \(=x^3-4x^2+13x-52\), Remainder \(=200\).
6) \((y^3-3y^2+5y-1)\div(y-1)\).
Quotient \(=y^2-2y+3\), Remainder \(=2\).
Practice Set 3.4 — Value of a Polynomial
1) \(x=0\) in \(x^2-5x+5\).
\(5\).
2) If \(p(y)=y^2-\dfrac{3}{2}y+1\), find \(p\!\left(\dfrac{3}{2}\right)\).
\(\left(\dfrac{3}{2}\right)^2-\dfrac{3}{2}\cdot\dfrac{3}{2}+1=\dfrac{9}{4}-\dfrac{9}{4}+1=1\).
3) If \(p(m)=m^3+2m^2-m+10\), find \(p(a)+p(-a)\).
\((a^3+2a^2-a+10)+(-a^3+2a^2+a+10)=4a^2+20\).
4) \(p(y)=2y^3-6y^2-5y+7\). Find \(p(2)\).
\(-11\).
Practice Set 3.5 — Remainder/Factor Theorem
1) Values of \(2x-2x^3+7\) for \(x=3,-1,0\).
At \(3\): \(6-54+7=-41\). At \(-1\): \(-2+2+7=7\). At \(0\): \(7\).
2) Find \(p(1),p(0),p(-2)\) for: (i) \(x^3\) (ii) \(y^2-2y+5\) (iii) \(x^4-2x^2-x\).
(i) \(1,0,-8\). (ii) \(4,5,13\). (iii) \(1,0,14\).
3) If \(m^3+2m+a=12\) at \(m=2\), find \(a\).
\(8+4+a=12\Rightarrow a=0\).
4) For \(p(x)=mx^2-2x+3\), if \(p(-1)=7\) find \(m\).
\(m(1)+2+3=7\Rightarrow m=2\).
5) Remainders by factor theorem: (i) \((x^2-7x+9)\) by \(x+1\) (ii) \((2x^3-2x^2+ax-a)\) by \(x-a\) (iii) \((54m^3+18m^2-27m+5)\) by \(m-3\).
(i) \(p(-1)=1+7+9=17\). (ii) \(p(a)=2a^3-2a^2+a^2-a=2a^3- a^2-a\). (iii) \(p(3)=54\cdot27+18\cdot9-27\cdot3+5=1458+162-81+5=1544\).
6) If \(y^3-5y^2+7y+m\) divided by \(y+2\) leaves remainder \(50\), find \(m\).
\(p(-2)=-8-20-14+m=50\Rightarrow m=92\).
7) Is \(x+3\) a factor of \(x^2+2x-3\)?
Check \(p(-3)=9-6-3=0\Rightarrow\) Yes.
8) If \(x-2\) is a factor of \(x^3-mx^2+10x-20\), find \(m\).
\(p(2)=8-4m+20-20=0\Rightarrow m=2\).
9) Decide factor: (i) \(x-1\) for \(x^3-x^2-x-1\) (ii) \(x-3\) for \(2x^3-x^2-45\).
(i) \(p(1)=1-1-1-1=-2\neq0\) ⇒ Not a factor. (ii) \(p(3)=54-9-45=0\) ⇒ Factor.
10) Remainder of \(x^{31}+31\) on division by \(x+1\).
\(p(-1)=-1+31=30\).
11) Show \(m-1\) is factor of \(m^{21}-1\) and \(m^{22}-1\).
Evaluate at \(m=1\): both give \(0\) ⇒ factor.
12*) If \(x-2\) and \(x-\dfrac12\) are factors of \(nx^2-5x+m\), show \(m=n=2\).
Use \(p(2)=0\Rightarrow4n-10+m=0\). And \(p(\frac12)=0\Rightarrow \frac{n}{4}-\frac{5}{2}+m=0\). Solve ⇒ \(n=2,\ m=2\).
13) (i) \(p(x)=2+5x\Rightarrow p(2)+p(-2)-p(1)\). (ii) \(p(x)=2x^2-\dfrac{5}{3}x+5\Rightarrow p\!\left(\dfrac{5}{3}\right)\).
(i) \((2+10)+(2-10)-(2+5)=12-8-7=-3\). (ii) \(2\cdot\frac{25}{9}-\frac{25}{9}+5=\frac{25}{9}+5=\frac{25+45}{9}=\frac{70}{9}\).
Practice Set 3.6 — Factors of Polynomials
1) Factor: (i) \(2x^2+x-1\) (ii) \(2m^2+5m-3\) (iii) \(12x^2+61x+77\) (iv) \(3y^2-2y-1\) (v) \(3x^2+4x+3\) (vi) \(\dfrac12x^2-3x+4\).
(i) \((2x-1)(x+1)\). (ii) \((2m-1)(m+3)\). (iii) \((4x+11)(3x+7)\). (iv) \((3y+1)(y-1)\). (v) Irreducible over \(\mathbb{Z}\) (discriminant \(<0\)). (vi) \(\tfrac12(x-2)(x-4)\).
2) Factorize: (i) \((x^2-x)^2-8(x^2-x)+12\) (ii) \((x-5)^2-(5x-25)-24\) (iii) \((x^2-6x)^2-8(x^2-6x+8)-64\) (iv) \((x^2-2x+3)(x^2-2x+5)-35\) (v) \((y+2)(y-3)(y+8)(y+3)+56\) (vi) \((y^2+5y)(y^2+5y-2)-24\) (vii) \((x-3)(x-4)^2(x-5)-6\).
(i) \((x-2)(x+1)(x-3)(x+2)\). (ii) \((x-13)(x-2)\). (iii) \((x-8)(x+2)(x-2)(x-4)\). (iv) \((x^2-2x+10)(x^2-2x-2)\). (v) \((y^2+5y-22)(y+1)(y+4)\). (vi) \((y+6)(y-1)(y+1)(y+4)\). (vii) Let \(t=x-4\): \(t^4-t^2-6=(t^2-3)(t^2+2)\) \(=\big((x-4)^2-3\big)\big((x-4)^2+2\big)\).
Problem Set 3 — Key Solutions (Concise)
1) MCQ quick answers: (i) Polynomial? (ii) Degree of \(7\)? (iii) Degree of zero polynomial? (iv) Degree of \(2x^2+5x^3+7\)? (v) Coefficient form of \(x^3-1\)? (x) Linear polynomial?
(i) \(2x^2+\tfrac12\). (ii) \(0\). (iii) Undefined. (iv) \(3\). (v) \((1,0,0,-1)\). (x) \(x+5\).
2) Degrees: (i) \(5+3x^4\) (ii) \(7\) (iii) \(ax^7+bx^9\).
\(4,\ 0,\ 9\) (if \(b\neq0\)).
3) Standard form: (i) \(4x^2+7x^4-x^3-x+9\) (ii) \(p+2p^3+10p^2+5p^4-8\).
\(7x^4-x^3+4x^2-x+9;\ 5p^4+2p^3+10p^2+p-8.\)
4) Coefficient forms: (i) \(x^4+16\) (ii) \(m^5+2m^2+3m+15\).
\((1,0,0,0,16);\ (1,0,2,3,15)\).
5) Index from coefficients: (i) \((3,-2,0,7,18)\) (ii) \((6,1,0,7)\).
\(3x^4-2x^3+0x^2+7x+18;\ 6x^3+x^2+0x+7.\)
6) Add: (i) \(7x^4-2x^3+x+10\) and \(3x^4+15x^3+9x^2-8x+2\) (ii) \(3p^3q+2p^2q+7\) and \(2p^2q+4pq-2p^3q\) (iii) \((4,5,-3,0)\).
(i) \(10x^4+13x^3+9x^2-7x+12\). (ii) \(p^3q+4p^2q+4pq+7\). (iii) Interprets to \(4x^3+5x^2-3x\).
7) Subtract: (i) \((5x^2-2y+9)-(3x^2+5y-7)\) (ii) \((2x^2+3x+5)-(x^2-2x+3)\).
(i) \(2x^2-7y+16\). (ii) \(x^2+5x+2\).
8) Multiply: (i) \((m^3-2m+3)(m^4-2m^2+3m+2)\) (ii) \((5m^3-2)(m^2-m+3)\).
(i) Expand systematically (highest degree \(7\)). (ii) \(5m^5-5m^4+15m^3-2m^2+2m-6\).
9) Divide \(3x^3-8x^2+x+7\) by \(x-3\) (synthetic). Quotient & remainder.
Coeffs \((3,-8,1,7)\) with \(3\): quotient \(=3x^2+x+4\), remainder \(=19\).
12) If remainders of \(bx^2+x+5\) and \(bx^3-2x+5\) on division by \(x-3\) are \(m,n\) with \(m-n=0\), find \(b\).
\(m=b(9)+3+5,\ n=b(27)-6+5\). Equality ⇒ \(9b+8=27b-1\Rightarrow b=\dfrac{9}{18}=\dfrac12\).
13) Simplify \((8m^2+3m-6)-(9m-7)+(3m^2-2m+4)\).
\(11m^2-8m+5\).
14) Which polynomial to subtract from \(x^2+13x+7\) to get \(3x^2+5x-4\)?
\((x^2+13x+7)-(\cdot)=3x^2+5x-4\Rightarrow\) subtract \((-2x^2+8x+11)\).
15) Which polynomial to add to \(4m+2n+3\) to get \(6m+3n+10\)?
Add \(2m+n+7\).
Mini Reference
• Degree of product = sum of degrees. • Remainder Theorem: remainder on \((x-a)\) is \(p(a)\).
• Factor Theorem: \(p(a)=0 \Rightarrow (x-a)\) is a factor. • Coefficient form lists all missing powers with 0.
Tip: Copy each part (1-mark, 2-mark, 3-mark, Exercises) separately so your site menu stays unchanged.