Factoring Polynomials Worksheets with Answer Key - Free Printable
Educational worksheet: Factoring Polynomials Worksheets with Answer Key. Download and print for classroom or home learning activities.
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Step-by-step solution for: Factoring Polynomials Worksheets with Answer Key
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Polynomials Worksheets with Answer Key
Here are the step-by-step solutions for each polynomial factoring problem.
1) $-3k^3 + 15k^2 - 6k$
* Step 1: Look for a Greatest Common Factor (GCF). All coefficients ($-3, 15, -6$) are divisible by $3$. Since the first term is negative, we factor out $-3$. All terms have at least one $k$. So, the GCF is $-3k$.
* Step 2: Divide each term by $-3k$:
* $-3k^3 \div -3k = k^2$
* $15k^2 \div -3k = -5k$
* $-6k \div -3k = 2$
* Step 3: Write the factored form: $-3k(k^2 - 5k + 2)$. The quadratic inside cannot be factored further using integers.
2) $64 + 40y^2 + 72y$
* Step 1: Rearrange into standard form: $40y^2 + 72y + 64$.
* Step 2: Find the GCF. $40, 72,$ and $64$ are all divisible by $8$.
* Step 3: Factor out $8$: $8(5y^2 + 9y + 8)$.
* Step 4: Check if $(5y^2 + 9y + 8)$ factors. We need two numbers that multiply to $40$ ($5 \times 8$) and add to $9$. No integer pairs work. So, it stays as is.
3) $ay + ax - yx - x^2$
* Step 1: This has 4 terms, so we use grouping. Group the first two and the last two: $(ay + ax) - (yx + x^2)$. Note: When grouping the second pair with a minus sign outside, the signs inside change if necessary, but here $-yx - x^2$ becomes $-(yx + x^2)$.
* Step 2: Factor out the GCF from each group.
* From $(ay + ax)$, factor out $a$: $a(y + x)$.
* From $-(yx + x^2)$, factor out $x$: $-x(y + x)$.
* Step 3: Now we have $a(y + x) - x(y + x)$. The common binomial is $(y + x)$.
* Step 4: Factor out $(y + x)$ to get $(y + x)(a - x)$.
4) $3x^3y + 18x^2y - 21xy$
* Step 1: Find the GCF. Coefficients $3, 18, 21$ are divisible by $3$. Variables: all have at least one $x$ and one $y$. GCF is $3xy$.
* Step 2: Divide each term by $3xy$:
* $3x^3y \div 3xy = x^2$
* $18x^2y \div 3xy = 6x$
* $-21xy \div 3xy = -7$
* Step 3: Write as $3xy(x^2 + 6x - 7)$.
* Step 4: Factor the quadratic $(x^2 + 6x - 7)$. We need numbers that multiply to $-7$ and add to $6$. These are $+7$ and $-1$.
* Step 5: Final form: $3xy(x + 7)(x - 1)$.
5) $3t^2 + 10t - 48$
* Step 1: This is a quadratic where $a \neq 1$. Multiply $a \cdot c$: $3 \cdot -48 = -144$.
* Step 2: Find two numbers that multiply to $-144$ and add to $b = 10$.
* Factors of $144$: $12 \cdot 12$, $18 \cdot 8$.
* Check signs: We need positive sum, so the larger number is positive. $+18$ and $-8$.
* $18 \cdot -8 = -144$ and $18 + (-8) = 10$.
* Step 3: Rewrite the middle term: $3t^2 + 18t - 8t - 48$.
* Step 4: Factor by grouping:
* $3t(t + 6) - 8(t + 6)$
* Step 5: Combine: $(3t - 8)(t + 6)$.
6) $x^2(x^2 - 5) + 6(x^2 - 5)$
* Step 1: Notice that $(x^2 - 5)$ is a common binomial factor in both parts.
* Step 2: Factor out $(x^2 - 5)$. You are left with $(x^2 + 6)$.
* Step 3: Result: $(x^2 - 5)(x^2 + 6)$. Neither of these can be factored further using integers.
7) $8a^2b^4 - 28a^3b^3 + 4a^2b^2$
* Step 1: Find the GCF.
* Coefficients: $8, 28, 4$. GCF is $4$.
* Variable $a$: Lowest power is $a^2$.
* Variable $b$: Lowest power is $b^2$.
* Total GCF: $4a^2b^2$.
* Step 2: Divide each term by $4a^2b^2$:
* $8a^2b^4 \div 4a^2b^2 = 2b^2$
* $-28a^3b^3 \div 4a^2b^2 = -7ab$
* $4a^2b^2 \div 4a^2b^2 = 1$
* Step 3: Write the result: $4a^2b^2(2b^2 - 7ab + 1)$. The inside part does not factor further.
8) $-90y^5 + 100y + 60$
* Step 1: Find the GCF. $90, 100, 60$ are all divisible by $10$. Since the leading term is negative, factor out $-10$. There is no common variable $y$ because the constant $60$ doesn't have one.
* Step 2: Factor out $-10$:
* $-90y^5 \div -10 = 9y^5$
* $100y \div -10 = -10y$
* $60 \div -10 = -6$
* Step 3: Result: $-10(9y^5 - 10y - 6)$. The polynomial inside cannot be easily factored further.
9) $2g^2 - 7g - 4$
* Step 1: Multiply $a \cdot c$: $2 \cdot -4 = -8$.
* Step 2: Find numbers that multiply to $-8$ and add to $-7$.
* Factors of $8$: $1, 8$.
* Signs: Need negative sum, so $-8$ and $+1$.
* $-8 + 1 = -7$.
* Step 3: Rewrite middle term: $2g^2 - 8g + 1g - 4$.
* Step 4: Group:
* $2g(g - 4) + 1(g - 4)$
* Step 5: Combine: $(2g + 1)(g - 4)$.
10) $y^3 + y^2 + 2y + 2$
* Step 1: Use grouping for 4 terms. Group first two and last two: $(y^3 + y^2) + (2y + 2)$.
* Step 2: Factor out GCF from each group.
* $(y^3 + y^2)$ becomes $y^2(y + 1)$.
* $(2y + 2)$ becomes $2(y + 1)$.
* Step 3: Now we have $y^2(y + 1) + 2(y + 1)$.
* Step 4: Factor out the common binomial $(y + 1)$.
* Step 5: Result: $(y + 1)(y^2 + 2)$.
Final Answer:
1) $-3k(k^2 - 5k + 2)$
2) $8(5y^2 + 9y + 8)$
3) $(y + x)(a - x)$
4) $3xy(x + 7)(x - 1)$
5) $(3t - 8)(t + 6)$
6) $(x^2 - 5)(x^2 + 6)$
7) $4a^2b^2(2b^2 - 7ab + 1)$
8) $-10(9y^5 - 10y - 6)$
9) $(2g + 1)(g - 4)$
10) $(y + 1)(y^2 + 2)$
1) $-3k^3 + 15k^2 - 6k$
* Step 1: Look for a Greatest Common Factor (GCF). All coefficients ($-3, 15, -6$) are divisible by $3$. Since the first term is negative, we factor out $-3$. All terms have at least one $k$. So, the GCF is $-3k$.
* Step 2: Divide each term by $-3k$:
* $-3k^3 \div -3k = k^2$
* $15k^2 \div -3k = -5k$
* $-6k \div -3k = 2$
* Step 3: Write the factored form: $-3k(k^2 - 5k + 2)$. The quadratic inside cannot be factored further using integers.
2) $64 + 40y^2 + 72y$
* Step 1: Rearrange into standard form: $40y^2 + 72y + 64$.
* Step 2: Find the GCF. $40, 72,$ and $64$ are all divisible by $8$.
* Step 3: Factor out $8$: $8(5y^2 + 9y + 8)$.
* Step 4: Check if $(5y^2 + 9y + 8)$ factors. We need two numbers that multiply to $40$ ($5 \times 8$) and add to $9$. No integer pairs work. So, it stays as is.
3) $ay + ax - yx - x^2$
* Step 1: This has 4 terms, so we use grouping. Group the first two and the last two: $(ay + ax) - (yx + x^2)$. Note: When grouping the second pair with a minus sign outside, the signs inside change if necessary, but here $-yx - x^2$ becomes $-(yx + x^2)$.
* Step 2: Factor out the GCF from each group.
* From $(ay + ax)$, factor out $a$: $a(y + x)$.
* From $-(yx + x^2)$, factor out $x$: $-x(y + x)$.
* Step 3: Now we have $a(y + x) - x(y + x)$. The common binomial is $(y + x)$.
* Step 4: Factor out $(y + x)$ to get $(y + x)(a - x)$.
4) $3x^3y + 18x^2y - 21xy$
* Step 1: Find the GCF. Coefficients $3, 18, 21$ are divisible by $3$. Variables: all have at least one $x$ and one $y$. GCF is $3xy$.
* Step 2: Divide each term by $3xy$:
* $3x^3y \div 3xy = x^2$
* $18x^2y \div 3xy = 6x$
* $-21xy \div 3xy = -7$
* Step 3: Write as $3xy(x^2 + 6x - 7)$.
* Step 4: Factor the quadratic $(x^2 + 6x - 7)$. We need numbers that multiply to $-7$ and add to $6$. These are $+7$ and $-1$.
* Step 5: Final form: $3xy(x + 7)(x - 1)$.
5) $3t^2 + 10t - 48$
* Step 1: This is a quadratic where $a \neq 1$. Multiply $a \cdot c$: $3 \cdot -48 = -144$.
* Step 2: Find two numbers that multiply to $-144$ and add to $b = 10$.
* Factors of $144$: $12 \cdot 12$, $18 \cdot 8$.
* Check signs: We need positive sum, so the larger number is positive. $+18$ and $-8$.
* $18 \cdot -8 = -144$ and $18 + (-8) = 10$.
* Step 3: Rewrite the middle term: $3t^2 + 18t - 8t - 48$.
* Step 4: Factor by grouping:
* $3t(t + 6) - 8(t + 6)$
* Step 5: Combine: $(3t - 8)(t + 6)$.
6) $x^2(x^2 - 5) + 6(x^2 - 5)$
* Step 1: Notice that $(x^2 - 5)$ is a common binomial factor in both parts.
* Step 2: Factor out $(x^2 - 5)$. You are left with $(x^2 + 6)$.
* Step 3: Result: $(x^2 - 5)(x^2 + 6)$. Neither of these can be factored further using integers.
7) $8a^2b^4 - 28a^3b^3 + 4a^2b^2$
* Step 1: Find the GCF.
* Coefficients: $8, 28, 4$. GCF is $4$.
* Variable $a$: Lowest power is $a^2$.
* Variable $b$: Lowest power is $b^2$.
* Total GCF: $4a^2b^2$.
* Step 2: Divide each term by $4a^2b^2$:
* $8a^2b^4 \div 4a^2b^2 = 2b^2$
* $-28a^3b^3 \div 4a^2b^2 = -7ab$
* $4a^2b^2 \div 4a^2b^2 = 1$
* Step 3: Write the result: $4a^2b^2(2b^2 - 7ab + 1)$. The inside part does not factor further.
8) $-90y^5 + 100y + 60$
* Step 1: Find the GCF. $90, 100, 60$ are all divisible by $10$. Since the leading term is negative, factor out $-10$. There is no common variable $y$ because the constant $60$ doesn't have one.
* Step 2: Factor out $-10$:
* $-90y^5 \div -10 = 9y^5$
* $100y \div -10 = -10y$
* $60 \div -10 = -6$
* Step 3: Result: $-10(9y^5 - 10y - 6)$. The polynomial inside cannot be easily factored further.
9) $2g^2 - 7g - 4$
* Step 1: Multiply $a \cdot c$: $2 \cdot -4 = -8$.
* Step 2: Find numbers that multiply to $-8$ and add to $-7$.
* Factors of $8$: $1, 8$.
* Signs: Need negative sum, so $-8$ and $+1$.
* $-8 + 1 = -7$.
* Step 3: Rewrite middle term: $2g^2 - 8g + 1g - 4$.
* Step 4: Group:
* $2g(g - 4) + 1(g - 4)$
* Step 5: Combine: $(2g + 1)(g - 4)$.
10) $y^3 + y^2 + 2y + 2$
* Step 1: Use grouping for 4 terms. Group first two and last two: $(y^3 + y^2) + (2y + 2)$.
* Step 2: Factor out GCF from each group.
* $(y^3 + y^2)$ becomes $y^2(y + 1)$.
* $(2y + 2)$ becomes $2(y + 1)$.
* Step 3: Now we have $y^2(y + 1) + 2(y + 1)$.
* Step 4: Factor out the common binomial $(y + 1)$.
* Step 5: Result: $(y + 1)(y^2 + 2)$.
Final Answer:
1) $-3k(k^2 - 5k + 2)$
2) $8(5y^2 + 9y + 8)$
3) $(y + x)(a - x)$
4) $3xy(x + 7)(x - 1)$
5) $(3t - 8)(t + 6)$
6) $(x^2 - 5)(x^2 + 6)$
7) $4a^2b^2(2b^2 - 7ab + 1)$
8) $-10(9y^5 - 10y - 6)$
9) $(2g + 1)(g - 4)$
10) $(y + 1)(y^2 + 2)$
Parent Tip: Review the logic above to help your child master the concept of factoring special products worksheet.
