Here are the step-by-step solutions for each problem on the worksheet.
1. $4x^2 + 4x - 35$
* Multiply $a \cdot c$: $4 \cdot -35 = -140$.
* Find factors of $-140$ that add up to $b$ ($4$). The numbers are $14$ and $-10$.
* Rewrite the middle term: $4x^2 + 14x - 10x - 35$.
* Factor by grouping: $2x(2x + 7) - 5(2x + 7)$.
* Combine: $(2x - 5)(2x + 7)$.
2. $5x^2 + 37x - 24$
* Multiply $a \cdot c$: $5 \cdot -24 = -120$.
* Find factors of $-120$ that add up to $37$. The numbers are $40$ and $-3$.
* Rewrite the middle term: $5x^2 + 40x - 3x - 24$.
* Factor by grouping: $5x(x + 8) - 3(x + 8)$.
* Combine: $(5x - 3)(x + 8)$.
3. $3x^2 + 12x + 12$
* First, look for a Greatest Common Factor (GCF). All terms are divisible by 3.
* Factor out 3: $3(x^2 + 4x + 4)$.
* Now factor the inside part. We need numbers that multiply to 4 and add to 4. Those are 2 and 2.
* Inside becomes: $(x + 2)(x + 2)$ or $(x + 2)^2$.
* Don't forget the 3 on the outside: $3(x + 2)(x + 2)$.
4. $6x^2 - 44x + 14$
* First, look for a GCF. All terms are even, so divide by 2.
* Factor out 2: $2(3x^2 - 22x + 7)$.
* Now factor the inside. Multiply $3 \cdot 7 = 21$. We need factors of 21 that add to $-22$. Those are $-21$ and $-1$.
* Rewrite middle term inside: $3x^2 - 21x - 1x + 7$.
* Group inside: $3x(x - 7) - 1(x - 7)$.
* Inside becomes: $(3x - 1)(x - 7)$.
* Add the 2 back: $2(3x - 1)(x - 7)$.
5. $3x^2 - 23x - 36$
* Multiply $a \cdot c$: $3 \cdot -36 = -108$.
* Find factors of $-108$ that add to $-23$. The numbers are $-27$ and $4$.
* Rewrite middle term: $3x^2 - 27x + 4x - 36$.
* Factor by grouping: $3x(x - 9) + 4(x - 9)$.
* Combine: $(3x + 4)(x - 9)$.
6. $2x^2 - 3x + 1$
* Multiply $a \cdot c$: $2 \cdot 1 = 2$.
* Find factors of $2$ that add to $-3$. The numbers are $-2$ and $-1$.
* Rewrite middle term: $2x^2 - 2x - 1x + 1$.
* Factor by grouping: $2x(x - 1) - 1(x - 1)$.
* Combine: $(2x - 1)(x - 1)$.
7. $20x^2 + 2x - 6$
* First, look for a GCF. All terms are even, so divide by 2.
* Factor out 2: $2(10x^2 + x - 3)$.
* Now factor the inside. Multiply $10 \cdot -3 = -30$. We need factors of $-30$ that add to $1$ (the invisible coefficient of $x$). Those are $6$ and $-5$.
* Rewrite middle term inside: $10x^2 + 6x - 5x - 3$.
* Group inside: $2x(5x + 3) - 1(5x + 3)$.
* Inside becomes: $(2x - 1)(5x + 3)$.
* Add the 2 back: $2(2x - 1)(5x + 3)$.
8. $3x^2 + 25x + 42$
* Multiply $a \cdot c$: $3 \cdot 42 = 126$.
* Find factors of $126$ that add to $25$. The numbers are $18$ and $7$.
* Rewrite middle term: $3x^2 + 18x + 7x + 42$.
* Factor by grouping: $3x(x + 6) + 7(x + 6)$.
* Combine: $(3x + 7)(x + 6)$.
9. $2x^2 - 3x - 20$
* Multiply $a \cdot c$: $2 \cdot -20 = -40$.
* Find factors of $-40$ that add to $-3$. The numbers are $-8$ and $5$.
* Rewrite middle term: $2x^2 - 8x + 5x - 20$.
* Factor by grouping: $2x(x - 4) + 5(x - 4)$.
* Combine: $(2x + 5)(x - 4)$.
10. $35x^2 - 11x - 6$
* Multiply $a \cdot c$: $35 \cdot -6 = -210$.
* Find factors of $-210$ that add to $-11$. The numbers are $-21$ and $10$.
* Rewrite middle term: $35x^2 - 21x + 10x - 6$.
* Factor by grouping: $7x(5x - 3) + 2(5x - 3)$.
* Combine: $(7x + 2)(5x - 3)$.
Final Answer:
1. $(2x - 5)(2x + 7)$
2. $(5x - 3)(x + 8)$
3. $3(x + 2)(x + 2)$
4. $2(3x - 1)(x - 7)$
5. $(3x + 4)(x - 9)$
6. $(2x - 1)(x - 1)$
7. $2(2x - 1)(5x + 3)$
8. $(3x + 7)(x + 6)$
9. $(2x + 5)(x - 4)$
10. $(7x + 2)(5x - 3)$
Parent Tip: Review the logic above to help your child master the concept of factoring polynomials practice worksheet.
