Factor p(x)=(x+2)(2x^2-7x+3)

Understand the Problem

The question is asking for the factorization of the polynomial function p(x) given as (x+2)(2x^2-7x+3). We will use polynomial multiplication and distribute to find the complete factorization.

Answer

The factorization of the polynomial function is $ p(x) = 2x^3 - 3x^2 - 11x + 6 $.

Steps to Solve

Expand the polynomial function

We start by multiplying the two binomials. We can use the distributive property (also known as the FOIL method for binomials). The expression is:

$$(x + 2)(2x^2 - 7x + 3)$$

To do this, we multiply each term in the first binomial by each term in the second polynomial.

Perform the distribution step by step

First, distribute $x$ across $2x^2 - 7x + 3$:

$$ x \cdot 2x^2 = 2x^3 $$

$$ x \cdot (-7x) = -7x^2 $$

$$ x \cdot 3 = 3x $$
Then, distribute $2$ across $2x^2 - 7x + 3$:

$$ 2 \cdot 2x^2 = 4x^2 $$

$$ 2 \cdot (-7x) = -14x $$

$$ 2 \cdot 3 = 6 $$

Combine all the terms

Now combine all the results:

$$ 2x^3 + (-7x^2 + 4x^2) + (3x - 14x) + 6 $$

This simplifies to:

$$ 2x^3 - 3x^2 - 11x + 6 $$

Final polynomial

Thus, the complete factorization of the polynomial is:

$$ p(x) = 2x^3 - 3x^2 - 11x + 6 $$

The factorization of the polynomial function is ( p(x) = 2x^3 - 3x^2 - 11x + 6 ).

More Information

The function ( p(x) ) is a cubic polynomial resulting from the multiplication of a linear and a quadratic polynomial. The factorization allows us to analyze the roots and behavior of the polynomial graphically.

Tips

Forgetting to multiply all terms in the second polynomial when using the distributive property.
Incorrectly combining like terms after distribution. To avoid this, carefully track each operation.

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