Polynomial Multiplication Worksheet - Free Printable

Let's solve each of these polynomial multiplication problems step by step. We'll use the distributive property (also known as the FOIL method for binomials) to expand each product.

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1) $(2r + 3)(7r + 7)$

Use distributive property:
$$
= 2r(7r) + 2r(7) + 3(7r) + 3(7)
= 14r^2 + 14r + 21r + 21
= 14r^2 + 35r + 21
$$

✔ Answer: $14r^2 + 35r + 21$

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2) $(8b + 6)(b - 5)$

$$
= 8b(b) + 8b(-5) + 6(b) + 6(-5)
= 8b^2 - 40b + 6b - 30
= 8b^2 - 34b - 30
$$

✔ Answer: $8b^2 - 34b - 30$

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3) $(7n + 5)(4n + 6)$

$$
= 7n(4n) + 7n(6) + 5(4n) + 5(6)
= 28n^2 + 42n + 20n + 30
= 28n^2 + 62n + 30
$$

✔ Answer: $28n^2 + 62n + 30$

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4) $(5x + 8)(6x + 3)$

$$
= 5x(6x) + 5x(3) + 8(6x) + 8(3)
= 30x^2 + 15x + 48x + 24
= 30x^2 + 63x + 24
$$

✔ Answer: $30x^2 + 63x + 24$

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5) $(a - 7)(2a + 2)$

$$
= a(2a) + a(2) + (-7)(2a) + (-7)(2)
= 2a^2 + 2a - 14a - 14
= 2a^2 - 12a - 14
$$

✔ Answer: $2a^2 - 12a - 14$

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6) $(4v + 8)(8v^2 - 3v - 4)$

Distribute each term in the first polynomial:

First: $4v(8v^2) = 32v^3$
$4v(-3v) = -12v^2$
$4v(-4) = -16v$

Second: $8(8v^2) = 64v^2$
$8(-3v) = -24v$
$8(-4) = -32$

Now combine:
$$
32v^3 -12v^2 -16v + 64v^2 -24v -32
= 32v^3 + ( -12v^2 + 64v^2 ) + ( -16v -24v ) -32
= 32v^3 + 52v^2 - 40v - 32
$$

✔ Answer: $32v^3 + 52v^2 - 40v - 32$

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7) $(x - 7)(5x^2 - 3x - 5)$

Distribute:

$x(5x^2) = 5x^3$
$x(-3x) = -3x^2$
$x(-5) = -5x$

$-7(5x^2) = -35x^2$
$-7(-3x) = 21x$
$-7(-5) = 35$

Combine:
$$
5x^3 -3x^2 -5x -35x^2 +21x +35
= 5x^3 + (-3x^2 -35x^2) + (-5x +21x) +35
= 5x^3 -38x^2 +16x +35
$$

✔ Answer: $5x^3 - 38x^2 + 16x + 35$

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8) $(6x - 4)(7x^2 + x - 4)$

Distribute:

$6x(7x^2) = 42x^3$
$6x(x) = 6x^2$
$6x(-4) = -24x$

$-4(7x^2) = -28x^2$
$-4(x) = -4x$
$-4(-4) = 16$

Combine:
$$
42x^3 + 6x^2 -24x -28x^2 -4x +16
= 42x^3 + (6x^2 -28x^2) + (-24x -4x) +16
= 42x^3 -22x^2 -28x +16
$$

✔ Answer: $42x^3 - 22x^2 - 28x + 16$

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9) $(3n - 2)(3n^2 - 8n - 5)$

Distribute:

$3n(3n^2) = 9n^3$
$3n(-8n) = -24n^2$
$3n(-5) = -15n$

$-2(3n^2) = -6n^2$
$-2(-8n) = 16n$
$-2(-5) = 10$

Combine:
$$
9n^3 -24n^2 -15n -6n^2 +16n +10
= 9n^3 + (-24n^2 -6n^2) + (-15n +16n) +10
= 9n^3 -30n^2 + n +10
$$

✔ Answer: $9n^3 - 30n^2 + n + 10$

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10) $(5k - 5)(k^2 - 4k - 5)$

Factor out 5 from first binomial: $5(k - 1)$, but we’ll just distribute directly.

$5k(k^2) = 5k^3$
$5k(-4k) = -20k^2$
$5k(-5) = -25k$

$-5(k^2) = -5k^2$
$-5(-4k) = 20k$
$-5(-5) = 25$

Combine:
$$
5k^3 -20k^2 -25k -5k^2 +20k +25
= 5k^3 + (-20k^2 -5k^2) + (-25k +20k) +25
= 5k^3 -25k^2 -5k +25
$$

✔ Answer: $5k^3 - 25k^2 - 5k + 25$

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11) $(2x^2 + 6x - 8)(2x^2 - 6x - 3)$

This is more complex — multiply each term in the first by each in the second.

We can factor both if possible, but let’s just expand:

Let’s do it term by term:

#### First: $2x^2(2x^2) = 4x^4$
$2x^2(-6x) = -12x^3$
$2x^2(-3) = -6x^2$

#### Second: $6x(2x^2) = 12x^3$
$6x(-6x) = -36x^2$
$6x(-3) = -18x$

#### Third: $-8(2x^2) = -16x^2$
$-8(-6x) = 48x$
$-8(-3) = 24$

Now sum all terms:

- $x^4$: $4x^4$
- $x^3$: $-12x^3 + 12x^3 = 0$
- $x^2$: $-6x^2 -36x^2 -16x^2 = -58x^2$
- $x$: $-18x + 48x = 30x$
- Constant: $24$

So:
$$
4x^4 - 58x^2 + 30x + 24
$$

✔ Answer: $4x^4 - 58x^2 + 30x + 24$

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12) $(2p^2 - 7p + 8)(2p^2 - 5p - 3)$

Again, distribute each term:

Let’s go term by term.

#### $2p^2(2p^2) = 4p^4$
$2p^2(-5p) = -10p^3$
$2p^2(-3) = -6p^2$

#### $-7p(2p^2) = -14p^3$
$-7p(-5p) = 35p^2$
$-7p(-3) = 21p$

#### $8(2p^2) = 16p^2$
$8(-5p) = -40p$
$8(-3) = -24$

Now combine:

- $p^4$: $4p^4$
- $p^3$: $-10p^3 -14p^3 = -24p^3$
- $p^2$: $-6p^2 +35p^2 +16p^2 = 45p^2$
- $p$: $21p -40p = -19p$
- Constant: $-24$

✔ Answer: $4p^4 - 24p^3 + 45p^2 - 19p - 24$

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13) $(5n^2 + 3n - 8)(4n^2 - 6n - 2)$

Multiply term by term.

#### $5n^2(4n^2) = 20n^4$
$5n^2(-6n) = -30n^3$
$5n^2(-2) = -10n^2$

#### $3n(4n^2) = 12n^3$
$3n(-6n) = -18n^2$
$3n(-2) = -6n$

#### $-8(4n^2) = -32n^2$
$-8(-6n) = 48n$
$-8(-2) = 16$

Now combine:

- $n^4$: $20n^4$
- $n^3$: $-30n^3 + 12n^3 = -18n^3$
- $n^2$: $-10n^2 -18n^2 -32n^2 = -60n^2$
- $n$: $-6n + 48n = 42n$
- Constant: $16$

✔ Answer: $20n^4 - 18n^3 - 60n^2 + 42n + 16$

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14) $(8m^2 + 8m + 3)(2m^2 + 8m + 4)$

Distribute:

#### $8m^2(2m^2) = 16m^4$
$8m^2(8m) = 64m^3$
$8m^2(4) = 32m^2$

#### $8m(2m^2) = 16m^3$
$8m(8m) = 64m^2$
$8m(4) = 32m$

#### $3(2m^2) = 6m^2$
$3(8m) = 24m$
$3(4) = 12$

Now combine:

- $m^4$: $16m^4$
- $m^3$: $64m^3 + 16m^3 = 80m^3$
- $m^2$: $32m^2 + 64m^2 + 6m^2 = 102m^2$
- $m$: $32m + 24m = 56m$
- Constant: $12$

✔ Answer: $16m^4 + 80m^3 + 102m^2 + 56m + 12$

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15) $(4x^2 - 7x - 1)(6x^2 + x - 7)$

Distribute:

#### $4x^2(6x^2) = 24x^4$
$4x^2(x) = 4x^3$
$4x^2(-7) = -28x^2$

#### $-7x(6x^2) = -42x^3$
$-7x(x) = -7x^2$
$-7x(-7) = 49x$

#### $-1(6x^2) = -6x^2$
$-1(x) = -x$
$-1(-7) = 7$

Now combine:

- $x^4$: $24x^4$
- $x^3$: $4x^3 - 42x^3 = -38x^3$
- $x^2$: $-28x^2 -7x^2 -6x^2 = -41x^2$
- $x$: $49x - x = 48x$
- Constant: $7$

✔ Answer: $24x^4 - 38x^3 - 41x^2 + 48x + 7$

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16) $(2r^2 + 5r - 1)(5r^2 + 2r - 8)$

Distribute:

#### $2r^2(5r^2) = 10r^4$
$2r^2(2r) = 4r^3$
$2r^2(-8) = -16r^2$

#### $5r(5r^2) = 25r^3$
$5r(2r) = 10r^2$
$5r(-8) = -40r$

#### $-1(5r^2) = -5r^2$
$-1(2r) = -2r$
$-1(-8) = 8$

Now combine:

- $r^4$: $10r^4$
- $r^3$: $4r^3 + 25r^3 = 29r^3$
- $r^2$: $-16r^2 + 10r^2 -5r^2 = -11r^2$
- $r$: $-40r -2r = -42r$
- Constant: $8$

✔ Answer: $10r^4 + 29r^3 - 11r^2 - 42r + 8$

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17) $(b^2 - 4b + 6)^2$

This is a square of a trinomial. Use:
$$
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
$$
Here: $a = b^2$, $b = -4b$, $c = 6$

So:
$$
= (b^2)^2 + (-4b)^2 + (6)^2 + 2(b^2)(-4b) + 2(b^2)(6) + 2(-4b)(6)
$$

Calculate:
- $b^4$
- $16b^2$
- $36$
- $2(b^2)(-4b) = -8b^3$
- $2(b^2)(6) = 12b^2$
- $2(-4b)(6) = -48b$

Now combine:
$$
b^4 -8b^3 + (16b^2 + 12b^2) -48b + 36
= b^4 -8b^3 + 28b^2 -48b + 36
$$

✔ Answer: $b^4 - 8b^3 + 28b^2 - 48b + 36$

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18) $(7n^2 + 8n + 7)(7n^2 + n - 5)$

Distribute:

#### $7n^2(7n^2) = 49n^4$
$7n^2(n) = 7n^3$
$7n^2(-5) = -35n^2$

#### $8n(7n^2) = 56n^3$
$8n(n) = 8n^2$
$8n(-5) = -40n$

#### $7(7n^2) = 49n^2$
$7(n) = 7n$
$7(-5) = -35$

Now combine:

- $n^4$: $49n^4$
- $n^3$: $7n^3 + 56n^3 = 63n^3$
- $n^2$: $-35n^2 + 8n^2 + 49n^2 = 22n^2$
- $n$: $-40n + 7n = -33n$
- Constant: $-35$

✔ Answer: $49n^4 + 63n^3 + 22n^2 - 33n - 35$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $14r^2 + 35r + 21$ |
| 2 | $8b^2 - 34b - 30$ |
| 3 | $28n^2 + 62n + 30$ |
| 4 | $30x^2 + 63x + 24$ |
| 5 | $2a^2 - 12a - 14$ |
| 6 | $32v^3 + 52v^2 - 40v - 32$ |
| 7 | $5x^3 - 38x^2 + 16x + 35$ |
| 8 | $42x^3 - 22x^2 - 28x + 16$ |
| 9 | $9n^3 - 30n^2 + n + 10$ |
| 10 | $5k^3 - 25k^2 - 5k + 25$ |
| 11 | $4x^4 - 58x^2 + 30x + 24$ |
| 12 | $4p^4 - 24p^3 + 45p^2 - 19p - 24$ |
| 13 | $20n^4 - 18n^3 - 60n^2 + 42n + 16$ |
| 14 | $16m^4 + 80m^3 + 102m^2 + 56m + 12$ |
| 15 | $24x^4 - 38x^3 - 41x^2 + 48x + 7$ |
| 16 | $10r^4 + 29r^3 - 11r^2 - 42r + 8$ |
| 17 | $b^4 - 8b^3 + 28b^2 - 48b + 36$ |
| 18 | $49n^4 + 63n^3 + 22n^2 - 33n - 35$ |

Let me know if you'd like this formatted as a printable worksheet or want explanations for any specific problem!