Problem 87 - A Graphical Approach to College Algebra

Question

Factor by any method. $$(4 t+5)^{2}+16(4 t+5)+64$$

Step-by-Step Solution

Answer

The expression factors to \((4t + 13)^2\).

Step 1: Identify the Structure

Notice that the expression \((4t+5)^2 + 16(4t+5) + 64\) has a quadratic form in terms of \((4t+5)\). Let \(x = 4t + 5\). This simplifies the expression to \(x^2 + 16x + 64\).

Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression \(x^2 + 16x + 64\). Look for two numbers that multiply to 64 and add to 16. These numbers are 8 and 8, so we can write the expression as \((x + 8)(x + 8)\) or \((x + 8)^2\).

Step 3: Substitute Back the Original Expression

Substitute \(x = 4t + 5\) back into the factored expression. This gives us \((4t + 5 + 8)^2\). Simplify this expression to get \((4t + 13)^2\).

Step 4: Final Expression Verification

The factored form of the original expression \((4t+5)^2 + 16(4t+5) + 64\) is indeed \((4t + 13)^2\). We have successfully factored the expression to confirm that this is accurate.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They usually have the general form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The expression we are dealing with here, \( (4t+5)^2 + 16(4t+5) + 64 \), is quadratic in nature due to its variable being squared.In this expression, we notice that \( (4t+5) \) itself acts as a single variable, allowing us to recognize it in the standard quadratic format \( x^2 + 16x + 64 \), where \( x = 4t + 5 \). Understanding this form is crucial since it helps identify the parts that need factoring. Decomposing complex polynomials like these into recognizable quadratic forms makes them easier to work with and analyze.

Substitution Method

The substitution method is a powerful tool in algebra that simplifies solving complex equations by temporarily replacing a more complicated part of the expression with a single variable. In the example we’re working through, substituting \( x = 4t + 5 \) converts our original expression into \( x^2 + 16x + 64 \).

• Identify the repetitive component or the one which makes the equation look cumbersome.
• Substitute this component with a simpler term or another variable. In this case: \( x = 4t + 5 \).
• Proceed to solve or factor the simplified expression.
• Once simplified, back-substitute the original terms in place of the variable.

Through this method, handling the expression becomes streamlined, allowing us to clearly identify patterns and factor them efficiently.

Polynomial Factoring

Polynomial factoring involves expressing a polynomial as a product of its factors. In quadratic expressions, such as \( x^2 + 16x + 64 \), factoring requires finding two binomials whose product gives the original polynomial. These binomials, when expanded, will include numbers that multiply to the constant term, 64, and add up to the linear coefficient, 16.In our example, since both required numbers are 8 (as \( 8 \times 8 = 64 \) and \( 8 + 8 = 16 \)), the expression can be factored into \((x + 8)(x + 8)\) or simplified to \((x + 8)^2\). This reflects the identity that squares the binomial: if \((a + b)^2\) is the form, it expands to \( a^2 + 2ab + b^2 \).Finally, replacing \( x \) back with \( 4t + 5 \) gives us \((4t + 13)^2\). Factoring polynomial expressions like these can be tricky, but by breaking them into manageable parts and using reliable factoring techniques, the task becomes significantly easier.