Multiplying a Polynomial and a Monomial worksheets - Free Printable

Let’s solve each problem step by step. These are all about multiplying a monomial (a single term) by a polynomial (an expression with multiple terms). We use the distributive property: multiply the outside term by *each* term inside the parentheses.

We’ll go one by one, carefully.

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1) $ x^2(x - 4) $

Multiply $x^2$ by $x$: → $x^3$
Multiply $x^2$ by $-4$: → $-4x^2$
→ Final: $x^3 - 4x^2$

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2) $ 3x(5x - 6) $

$3x \cdot 5x = 15x^2$
$3x \cdot (-6) = -18x$
→ Final: $15x^2 - 18x$

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

$3 \cdot x^2 = 3x^2$
$3 \cdot (-2x) = -6x$
→ Final: $3x^2 - 6x$

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4) $ -x(x - 3) $

$-x \cdot x = -x^2$
$-x \cdot (-3) = +3x$
→ Final: $-x^2 + 3x$

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5) $ 5x(2x^2 - 4x + 5) $

$5x \cdot 2x^2 = 10x^3$
$5x \cdot (-4x) = -20x^2$
$5x \cdot 5 = 25x$
→ Final: $10x^3 - 20x^2 + 25x$

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6) $ x(2x + 5y) $

$x \cdot 2x = 2x^2$
$x \cdot 5y = 5xy$
→ Final: $2x^2 + 5xy$

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7) $ 7x(3x - 6y) $

$7x \cdot 3x = 21x^2$
$7x \cdot (-6y) = -42xy$
→ Final: $21x^2 - 42xy$

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8) $ 2xy(2x - 3) $

$2xy \cdot 2x = 4x^2y$
$2xy \cdot (-3) = -6xy$
→ Final: $4x^2y - 6xy$

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9) $ x^2y(2y + 1) $

$x^2y \cdot 2y = 2x^2y^2$
$x^2y \cdot 1 = x^2y$
→ Final: $2x^2y^2 + x^2y$

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10) $ 5ab(3a - 5b) $

$5ab \cdot 3a = 15a^2b$
$5ab \cdot (-5b) = -25ab^2$
→ Final: $15a^2b - 25ab^2$

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11) $ 6ac(7a - 5c + 4) $

$6ac \cdot 7a = 42a^2c$
$6ac \cdot (-5c) = -30ac^2$
$6ac \cdot 4 = 24ac$
→ Final: $42a^2c - 30ac^2 + 24ac$

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12) $ b^2(3a - 25b + 6) $

$b^2 \cdot 3a = 3ab^2$
$b^2 \cdot (-25b) = -25b^3$
$b^2 \cdot 6 = 6b^2$
→ Final: $3ab^2 - 25b^3 + 6b^2$

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13) $ x^2y^2(5x^2y - 3xy^2) $

$x^2y^2 \cdot 5x^2y = 5x^{4}y^{3}$
$x^2y^2 \cdot (-3xy^2) = -3x^{3}y^{4}$
→ Final: $5x^4y^3 - 3x^3y^4$

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14) $ 5x^3y^2(2xy + 3x^2y^2) $

$5x^3y^2 \cdot 2xy = 10x^4y^3$
$5x^3y^2 \cdot 3x^2y^2 = 15x^5y^4$
→ Final: $10x^4y^3 + 15x^5y^4$

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15) $ 3a^3b(2a - 9b) $

$3a^3b \cdot 2a = 6a^4b$
$3a^3b \cdot (-9b) = -27a^3b^2$
→ Final: $6a^4b - 27a^3b^2$

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16) $ 12abc(a^2 + b^2 - c^2) $

$12abc \cdot a^2 = 12a^3bc$
$12abc \cdot b^2 = 12ab^3c$
$12abc \cdot (-c^2) = -12abc^3$
→ Final: $12a^3bc + 12ab^3c - 12abc^3$

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17) $ x^5y^3(-2x^2 + 4y) $

$x^5y^3 \cdot (-2x^2) = -2x^7y^3$
$x^5y^3 \cdot 4y = 4x^5y^4$
→ Final: $-2x^7y^3 + 4x^5y^4$

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18) $ x^4y^3z(2x^2 - 3zy + 5y) $

Note: “zy” is same as “yz”, but we’ll keep order as given unless specified. Let’s write variables in alphabetical order for consistency later if needed — but here, let’s just compute:

$x^4y^3z \cdot 2x^2 = 2x^6y^3z$
$x^4y^3z \cdot (-3zy) = -3x^4y^4z^2$ （because z·z = z², y³·y = y⁴）
$x^4y^3z \cdot 5y = 5x^4y^4z$
→ Final: $2x^6y^3z - 3x^4y^4z^2 + 5x^4y^4z$

*(You can rearrange terms if you want, but this is correct as is.)*

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19) $ x^3y^2(2xy^2 - 7xy + 3) $

$x^3y^2 \cdot 2xy^2 = 2x^4y^4$
$x^3y^2 \cdot (-7xy) = -7x^4y^3$
$x^3y^2 \cdot 3 = 3x^3y^2$
→ Final: $2x^4y^4 - 7x^4y^3 + 3x^3y^2$

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20) $ x^4y(7x + 9xy - 6x^3y^5) $

$x^4y \cdot 7x = 7x^5y$
$x^4y \cdot 9xy = 9x^5y^2$
$x^4y \cdot (-6x^3y^5) = -6x^7y^6$
→ Final: $7x^5y + 9x^5y^2 - 6x^7y^6$

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21) $ 5xy^2(2z + 5x^2y + 2xy^2z) $

$5xy^2 \cdot 2z = 10xy^2z$
$5xy^2 \cdot 5x^2y = 25x^3y^3$
$5xy^2 \cdot 2xy^2z = 10x^2y^4z$
→ Final: $10xy^2z + 25x^3y^3 + 10x^2y^4z$

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22) $ x^3y^2z^4(7xyz^3 + x^2z - 6xzy^3) $

First, note: “xzy” = “xyz”, so last term is $-6xyz y^3 = -6xy^4z$, but let’s compute directly:

Term 1: $x^3y^2z^4 \cdot 7xyz^3 = 7x^{4}y^{3}z^{7}$
Term 2: $x^3y^2z^4 \cdot x^2z = x^{5}y^{2}z^{5}$
Term 3: $x^3y^2z^4 \cdot (-6xzy^3) = -6x^{4}y^{5}z^{5}$ （since x·x = x²? Wait — no: x³ · x = x⁴; y² · y³ = y⁵; z · z = z⁵）

Wait — let me double-check Term 3:

Original: $-6xzy^3$ → that’s $-6 \cdot x \cdot z \cdot y^3$

So:
$x^3y^2z^4 \cdot (-6x z y^3) = -6 \cdot x^{3+1} \cdot y^{2+3} \cdot z^{4+1} = -6x^4y^5z^5$

Yes.

So final answer:

→ $7x^4y^3z^7 + x^5y^2z^5 - 6x^4y^5z^5$

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✔ All problems solved using distributive property and exponent rules (add exponents when multiplying same base).

Final Answer:
1) $x^3 - 4x^2$
2) $15x^2 - 18x$
3) $3x^2 - 6x$
4) $-x^2 + 3x$
5) $10x^3 - 20x^2 + 25x$
6) $2x^2 + 5xy$
7) $21x^2 - 42xy$
8) $4x^2y - 6xy$
9) $2x^2y^2 + x^2y$
10) $15a^2b - 25ab^2$
11) $42a^2c - 30ac^2 + 24ac$
12) $3ab^2 - 25b^3 + 6b^2$
13) $5x^4y^3 - 3x^3y^4$
14) $10x^4y^3 + 15x^5y^4$
15) $6a^4b - 27a^3b^2$
16) $12a^3bc + 12ab^3c - 12abc^3$
17) $-2x^7y^3 + 4x^5y^4$
18) $2x^6y^3z - 3x^4y^4z^2 + 5x^4y^4z$
19) $2x^4y^4 - 7x^4y^3 + 3x^3y^2$
20) $7x^5y + 9x^5y^2 - 6x^7y^6$
21) $10xy^2z + 25x^3y^3 + 10x^2y^4z$
22) $7x^4y^3z^7 + x^5y^2z^5 - 6x^4y^5z^5$