Question

which statements are true about the polynomial $4x^3 - 6x^2 + 8x - 12$? check all that apply.
the terms $4x^3$ and $8x$ have a common factor.
the terms $4x^3$ and $-6x^2$ have a common factor.
the polynomial is prime.
the factored polynomial is $(2x^2 - 3)(2x + 4)$.
the polynomial can be grouped in different ways to factor by grouping.

Explanation:

Step1: Check common factor of $4x^3,8x$

Factor terms: $4x^3=4x\cdot x^2$, $8x=4x\cdot2$. Common factor: $4x$.

Step2: Check common factor of $4x^3,-6x^2$

Factor terms: $4x^3=2x^2\cdot2$, $-6x^2=2x^2\cdot(-3)$. Common factor: $2x^2$.

Step3: Check if polynomial is prime

Factor by grouping:
Group 1: $4x^3-6x^2=2x^2(2x-3)$
Group 2: $8x-12=4(2x-3)$
Combine: $(2x^2+4)(2x-3)=2(x^2+2)(2x-3)$. Not prime.

Step4: Verify $(2x^2-3)(2x+4)$

Expand: $(2x^2)(2x)+(2x^2)(4)-3(2x)-3(4)=4x^3+8x^2-6x-12$, not equal to original polynomial.

Step5: Check grouping flexibility

Can group as $(4x^3+8x)+(-6x^2-12)=4x(x^2+2)-6(x^2+2)=(4x-6)(x^2+2)=2(2x-3)(x^2+2)$, so multiple grouping ways work.

Answer:

• The terms $4x^3$ and $8x$ have a common factor.
• The terms $4x^3$ and $-6x^2$ have a common factor.
• The polynomial can be grouped in different ways to factor by grouping.