Problem 33 - Elementary Algebra

Question

Factor. $$ 16 x 2+26 x-10 $$

Step-by-Step Solution

Answer

The expression factors to \(2(4x - 1)(2x + 5)\).

Step 1: Identify Coefficients

To factor the quadratic expression \(16x^2 + 26x - 10\), first identify the coefficients: \(a = 16\), \(b = 26\), and \(c = -10\).

Step 2: Multiply and Find Product

Multiply the leading coefficient \(a\) by the constant term \(c\): \(16 \times (-10) = -160\). We will find two numbers that multiply to \(-160\) and add to \(26\).

Step 3: Determine Factor Pair

Consider factor pairs of \(-160\): Find a pair that sums to \(26\). The pair \(40\) and \(-4\) works: \(40 + (-4) = 36\) and \(40 \times (-4) = -160\).

Step 4: Rewrite the Middle Term

Rewrite the equation by splitting the middle term into the two numbers found: \(16x^2 + 40x - 4x - 10\).

Step 5: Group Terms

Group the terms into two pairs: \((16x^2 + 40x) + (-4x - 10)\).

Step 6: Factor by Grouping

Factor the greatest common factor from each group: \(8x(2x + 5) - 2(2x + 5)\).

Step 7: Factor Out Common Binomial

Factor out the common binomial \((2x + 5)\): \((8x - 2)(2x + 5)\).

Step 8: Simplify

Further simplify \(8x - 2\) by factoring out 2: \(2(4x - 1)(2x + 5)\). Therefore, the factorization is \(2(4x - 1)(2x + 5)\).

Key Concepts

Quadratic Expression

A quadratic expression is an algebraic expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). It is called quadratic because the highest power of the variable \( x \) is 2, which is known as the quadratic term.
Understanding the structure of a quadratic expression is crucial, as it helps in determining the right approach for solving or factoring these expressions.
Quadratic expressions appear in various mathematical and real-world problems, like calculating areas, designing physics experiments, and more.

Coefficients

Coefficients are the numerical factors of the terms in an expression. In the quadratic expression \( 16x^2 + 26x - 10 \), each term has a coefficient:

• The coefficient of \( x^2 \) is \( 16 \), known as the leading coefficient.
• The coefficient of \( x \) is \( 26 \).
• The constant term is \(-10 \).

Coefficients are crucial as they affect the shape and position of the graph when plotted. In factoring, knowing these helps identify the factor pairs necessary for simplifying the expression.

Factor by Grouping

Factoring by grouping is a method used to factor expressions that consist of four or more terms. It involves rearranging and grouping the terms so that common factors can be factored out, ultimately simplifying the expression.
For example, in the expression \( 16x^2 + 40x - 4x - 10 \), we group the terms as \((16x^2 + 40x) + (-4x - 10)\). Then, we factor out the GCF from each group:

• From \( 16x^2 + 40x \), common factor is \( 8x \).
• From \(-4x - 10 \), common factor is \(-2 \).

This step is vital because it sets the stage for factoring out a common binomial, which simplifies the expression entirely.

Polynomials

Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative whole number exponents. A quadratic expression like \( 16x^2 + 26x - 10 \) is a simple form of a polynomial.
Polynomials are classified by their degree and the number of terms they have. Quadratic polynomials, like the example, have a degree of 2 due to the \( x^2 \) term.
Understanding polynomials is critical as it underlies almost all of algebra and extends into calculus and beyond. Recognizing the form and behavior of polynomials allows us to apply strategies like factoring, which simplifies solving for roots or zeros.