Part 1: Add, subtract, and multiply polynomial expressions
1) Rewrite as an equivalent expression in simplest form: \( 3x^4-2x^2-5-(2x^4+6x^3-3x^2+4x)\)
Tim thinks the answer is \(x^4+6x^3-5x^2+4x-5\)
Nick thinks the answer is \(x^4-6x^3+x^2-4x-5\)
Decide whose answer is correct, and explain why.
2) Simplify the following expression: \((2x+5)^3\)
Rachel thinks the answer is \(8x^3+60x^2+150x+125\)
Tina thinks the answer is \(8x^3+125\)
Decide whose answer is correct, and explain why.
Simplify the following expressions:
3) \((2x^3-5x^2+4x)+(5x^3-2x^2-3x-7)\)
4) \((15x^3-7x^2+8x)-(4x^3+10x-5)\)
5) \((-x^2+12x-6)+(9x^3-3x+2)\)
6) \((5x^2-14x+1)-4(2x^2+8x-9)\)
7) \((x-4)^3\)
8) \((x^2-6x+2)(3x+1)\)
9) \((5x+3)^2-2(x-4)^2\)
10) \((2x-1)(4x^2+2x+1)\)
11) \((x+6)^2(2x-3)\)
Use the following functions to complete problems 12-14.
\(f(x)=3x^2-4x+7\) \(g(x)=-5x^2-4x+1\) \(h(x)=8x+3\)
12) \(f(x)+h(x)\)
13) \(f(x)-g(x)\)
14) \(g(x)\cdot h(x)\)
Part 2: Divide Polynomial Expressions
15) Divide \(3x^3-11x^2-2x+24\) by \(3x+4\)
16) Divide \(10x^3-32x^2+21x-3\) by \(5x-1\)
17) Divide \(4x^4-8x^3-5x^2+4x+1\) by \(2x+3\)
18) Divide \(8x^4-10x^2+12x+4\) by \(4x-6\)
19) Divide \(3x^3-17x^2+14x-20\) by \(x-5\)
20) Divide \(2x^3+21x^2+46x-31\) by \(x+6\)
21) Divide \(2x^4-15x^2+1\) by \(x-3\)
Part 3: Factor Polynomial Expressions
Factor the following expressions:
22) \(x^4-8x^2+12\)
23) \(x^4-64\)
24) \(2x^6+5x^3-3\)
25) \(2x^5-50x\)
26) \(4x^2+49\)
27) \(x^4-18x^2-40\)
28) \(x^6+x^3-30\)
29) \(x^3+100x\)
30) \(x^3-64\)
31) \(27x^3+125\)
32) \(2x^4-32\)
33) \(5x^4+13x^2-6\)
34) \(12x^4-7x^2+1\)
35) \(x^5+2x^3-24x\)
36) \(16x^3-2\)
For problems 37-39 factor the expressions by grouping:
37) \(2x^3+3x^2-10x-15\)
38) \(3x^3+2x^2-27x-18\)
39) \(2x^3-x^2+50x-25\)
40) Factor \(x^3+3x^2-18x-40\) completely, given that \((x-4)\) is a factor.
41) Factor \(x^4+5x^3-8x-40\) completely, given that \((x+5)\) is a factor.
42) Factor \(x^5+2x^4-6x^3-12x^2+9x+18\) completely, given that \((x+2)\) is a factor.
43) Factor \(6x^3-5x^2-3x+2\) completely, given that \((3x+2)\) is a factor.
Solution Bank
1) Rewrite as an equivalent expression in simplest form: \( 3x^4-2x^2-5-(2x^4+6x^3-3x^2+4x)\)
Tim thinks the answer is \(x^4+6x^3-5x^2+4x-5\)
Nick thinks the answer is \(x^4-6x^3+x^2-4x-5\)
Decide whose answer is correct, and explain why.
2) Simplify the following expression: \((2x+5)^3\)
Rachel thinks the answer is \(8x^3+60x^2+150x+125\)
Tina thinks the answer is \(8x^3+125\)
Decide whose answer is correct, and explain why.
Simplify the following expressions:
3) \((2x^3-5x^2+4x)+(5x^3-2x^2-3x-7)\)
4) \((15x^3-7x^2+8x)-(4x^3+10x-5)\)
5) \((-x^2+12x-6)+(9x^3-3x+2)\)
6) \((5x^2-14x+1)-4(2x^2+8x-9)\)
7) \((x-4)^3\)
8) \((x^2-6x+2)(3x+1)\)
9) \((5x+3)^2-2(x-4)^2\)
10) \((2x-1)(4x^2+2x+1)\)
11) \((x+6)^2(2x-3)\)
Use the following functions to complete problems 12-14.
\(f(x)=3x^2-4x+7\) \(g(x)=-5x^2-4x+1\) \(h(x)=8x+3\)
12) \(f(x)+h(x)\)
13) \(f(x)-g(x)\)
14) \(g(x)\cdot h(x)\)
Part 2: Divide Polynomial Expressions
15) Divide \(3x^3-11x^2-2x+24\) by \(3x+4\)
16) Divide \(10x^3-32x^2+21x-3\) by \(5x-1\)
17) Divide \(4x^4-8x^3-5x^2+4x+1\) by \(2x+3\)
18) Divide \(8x^4-10x^2+12x+4\) by \(4x-6\)
19) Divide \(3x^3-17x^2+14x-20\) by \(x-5\)
20) Divide \(2x^3+21x^2+46x-31\) by \(x+6\)
21) Divide \(2x^4-15x^2+1\) by \(x-3\)
Part 3: Factor Polynomial Expressions
Factor the following expressions:
22) \(x^4-8x^2+12\)
23) \(x^4-64\)
24) \(2x^6+5x^3-3\)
25) \(2x^5-50x\)
26) \(4x^2+49\)
27) \(x^4-18x^2-40\)
28) \(x^6+x^3-30\)
29) \(x^3+100x\)
30) \(x^3-64\)
31) \(27x^3+125\)
32) \(2x^4-32\)
33) \(5x^4+13x^2-6\)
34) \(12x^4-7x^2+1\)
35) \(x^5+2x^3-24x\)
36) \(16x^3-2\)
For problems 37-39 factor the expressions by grouping:
37) \(2x^3+3x^2-10x-15\)
38) \(3x^3+2x^2-27x-18\)
39) \(2x^3-x^2+50x-25\)
40) Factor \(x^3+3x^2-18x-40\) completely, given that \((x-4)\) is a factor.
41) Factor \(x^4+5x^3-8x-40\) completely, given that \((x+5)\) is a factor.
42) Factor \(x^5+2x^4-6x^3-12x^2+9x+18\) completely, given that \((x+2)\) is a factor.
43) Factor \(6x^3-5x^2-3x+2\) completely, given that \((3x+2)\) is a factor.
Solution Bank