Question

Factor each polynomial. $$ 5 x^{2}-6 x-8 $$

Short answer

\((x - 2)(5x + 4)\)

Step-by-step solution

- Identify the Polynomial

The given polynomial is \(5x^2 - 6x - 8\). The goal is to factor this quadratic expression.

- Use the AC Method

For factoring by grouping, consider the polynomial in the form \(ax^2 + bx + c\). Here, \(a = 5\), \(b = -6\), and \(c = -8\). Find two numbers that multiply to \(a \times c = 5 \times (-8) = -40\) and add up to \(b = -6\). These numbers are \(-10\) and \(4\) because \(-10 \times 4 = -40\) and \(-10 + 4 = -6\).

- Rewrite the Middle Term

Rewrite the polynomial by splitting the middle term using the numbers found: \(5x^2 - 10x + 4x - 8\).

- Factor by Grouping

Group the terms in pairs and factor out the common factors: \(5x(x - 2) + 4(x - 2)\).

- Factor the Common Binomial

Factor out the common binomial \((x - 2)\): \((x - 2)(5x + 4)\).

- Verify the Answer

Expand the factors to check: \((x - 2)(5x + 4) = 5x^2 + 4x - 10x - 8 = 5x^2 - 6x - 8\). The factored form is correct.

Key concepts

Quadratic Expressions

A quadratic expression has the general form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The term quadratic refers to the highest power of the variable being squared. In the expression provided, \( 5x^2 - 6x - 8 \), you can see that it's a quadratic expression since the highest power of \( x \) is 2.
Understanding the structure of a quadratic expression helps in identifying the suitable method for factoring it. For instance, recognizing that the expression is in standard form \( ax^2 + bx + c \) is crucial before applying methods like factoring by grouping or the AC method. Remember, quadratic expressions are fundamental in algebra, and mastering them lays a strong foundation for solving more complex polynomial equations.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms in a polynomial to make it easier to factor. This method is particularly effective when dealing with four-term polynomials or when using the AC method.
In our example, after applying the AC method in the previous step, we rewrite the expression \( 5x^2 - 10x + 4x - 8 \). Notice how the middle term, \( -6x \), is broken into \( -10x \) and \( 4x \) based on the product-sum relationship identified. Now, we group the terms into pairs: \( (5x^2 - 10x) + (4x - 8) \).
Next, we factor out the common factor from each pair. For the first group, \( 5x^2 - 10x \), the common factor is \( 5x \), resulting in \( 5x(x - 2) \). For the second group, \( 4x - 8 \), the common factor is \( 4 \), resulting in \( 4(x - 2) \). Finally, we factor out the common binomial \( (x - 2) \) from both groups, resulting in the factored expression \( (x - 2)(5x + 4) \).

AC Method

The AC method is a systematic approach to factor quadratic expressions of the form \( ax^2 + bx + c \) when \( a eq 1 \).
This method involves multiplying the coefficient of the quadratic term (\( a \)) by the constant term (\( c \)), and identifying two numbers that multiply to this product (\( ac \)) and add up to the linear term coefficient (\( b \)). Let's look into our example in detail to illustrate this process:

• Given the quadratic expression \( 5x^2 - 6x - 8 \), we have \( a = 5 \), \( b = -6 \), and \( c = -8 \).
• The product \( ac = 5 \times (-8) = -40 \).
• We need two numbers that multiply to \( -40 \) and add up to \( -6 \). These numbers are \( -10 \) and \( 4 \), since \( -10 \times 4 = -40 \) and \( -10 + 4 = -6 \).

Now, we rewrite the quadratic expression by splitting the middle term using these two numbers, as follows: \( 5x^2 - 10x + 4x - 8 \). This prepares the expression for factoring by grouping.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials, which when multiplied together give the original polynomial. This is a crucial skill in algebra as it simplifies expressions and solves polynomial equations.
In our example, the quadratic expression \( 5x^2 - 6x - 8 \) was factored using the AC method followed by factoring by grouping. The final factored form is \( (x - 2)(5x + 4) \).
Verification is important to ensure the correctness of factorization. To verify, we multiply the factors back together:
\( (x - 2)(5x + 4) = 5x(x - 2) + 4(x - 2) = 5x^2 + 4x - 10x - 8 = 5x^2 - 6x - 8 \).
Remember, successful polynomial factorization often depends on choosing the right method and carefully checking each step for accuracy. Practicing these types of problems will help you become more proficient over time.