Question

use the gcf to factor each polynomial.

1. 14y² - 8
2. 6y⁴ - 18y² - 24y
3. 16x⁴ - 8x³ + 2x²
4. 32x⁴ + 24x³ + 16x²
5. write an expression in simplest form for the area of the shaded region. area - multiply length x width
6. write an expression in simplest form for the perimeter of the triangle. perimeter - add up the sides

Explanation:

Step1: Factor 14$y^{2}-8$

Find GCF of 14 and 8. Prime - factorize: $14 = 2\times7$, $8 = 2\times2\times2$. GCF of 14 and 8 is 2. So, $14y^{2}-8=2(7y^{2}-4)$.

Step2: Factor $6y^{4}-18y^{2}-24y$

Find GCF of 6, 18 and 24. Prime - factorize: $6 = 2\times3$, $18 = 2\times3\times3$, $24 = 2\times2\times2\times3$. GCF is 6. Also, each term has a factor of $y$. So, $6y^{4}-18y^{2}-24y = 6y(y^{3}-3y - 4)$.

Step3: Factor $16x^{4}-8x^{3}+2x^{2}$

Find GCF of 16, 8 and 2. Prime - factorize: $16 = 2\times2\times2\times2$, $8 = 2\times2\times2$, $2 = 2$. GCF is 2. Each term has a factor of $x^{2}$. So, $16x^{4}-8x^{3}+2x^{2}=2x^{2}(8x^{2}-4x + 1)$.

Step4: Factor $32x^{5}+24x^{3}+16x^{2}$

Find GCF of 32, 24 and 16. Prime - factorize: $32 = 2\times2\times2\times2\times2$, $24 = 2\times2\times2\times3$, $16 = 2\times2\times2\times2$. GCF is 8. Each term has a factor of $x^{2}$. So, $32x^{5}+24x^{3}+16x^{2}=8x^{2}(4x^{3}+3x + 2)$.

Step5: Find area of shaded region

Area of large rectangle is $(2x - 3)(x + 8)=2x^{2}+16x-3x - 24=2x^{2}+13x - 24$. Area of small square is $(x + 4)^{2}=x^{2}+8x + 16$. Shaded - area $A=(2x^{2}+13x - 24)-(x^{2}+8x + 16)=2x^{2}+13x - 24 - x^{2}-8x - 16=x^{2}+5x - 40$.

Step6: Find perimeter of triangle

Perimeter $P=(6x - 7)+(6x - 7)+2x=6x-7+6x - 7+2x=14x-14$.

Answer:

1. $2(7y^{2}-4)$
2. $6y(y^{3}-3y - 4)$
3. $2x^{2}(8x^{2}-4x + 1)$
4. $8x^{2}(4x^{3}+3x + 2)$
5. $x^{2}+5x - 40$
6. $14x-14$