Factor each polynomial completely. See Example 6. 4x² - 28x + 40

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First, identify the greatest common factor (GCF) of all the terms in the polynomial \(4x^{2} - 28x + 40\). The GCF is the largest number that divides each coefficient evenly.

Factor out the GCF from the polynomial. This means rewriting the polynomial as \(4(x^{2} - 7x + 10)\).

Next, focus on factoring the quadratic inside the parentheses: \(x^{2} - 7x + 10\). Look for two numbers that multiply to \(10\) (the constant term) and add up to \(-7\) (the coefficient of \(x\)).

Write the quadratic as a product of two binomials using the two numbers found: \((x - a)(x - b)\), where \(a\) and \(b\) are the numbers from the previous step.

Finally, write the completely factored form by combining the GCF and the factored quadratic: \(4(x - a)(x - b)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Quadratic Polynomials

Factoring quadratic polynomials involves expressing a quadratic expression in the form ax² + bx + c as a product of two binomials. This process helps simplify expressions and solve equations by finding the roots of the polynomial.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest factor that divides all terms of a polynomial. Factoring out the GCF first simplifies the polynomial, making it easier to factor the remaining quadratic expression.

Factoring by Grouping and Using the AC Method

The AC method involves multiplying the coefficient of x² (a) and the constant term (c), then finding two numbers that multiply to ac and add to b. This helps split the middle term and factor by grouping, breaking the polynomial into simpler binomials.

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