Problem 79 - Algebra 2

Question

Write each function in factored form. Check by multiplying. $$ y=4 x^{3}-49 x $$

Step-by-Step Solution

Answer

The function \(y = 4x^{3} - 49x\) can be factored to \(y = x(2x + 7)(2x - 7)\). After checking through multiplication, the original function is confirmed.

Step 1: Identify Common Factors

A common factor is a number or variable that divides evenly into all terms. Looking at the function \(y=4x^{3}-49x\), it can be seen that 'x' is a common factor. As such, 'x' can be factored out to give: \(y = x(4x^{2} - 49)\)

Step 2: Apply Difference of Squares Formula

The expression within the parentheses is now a difference of squares, and can be factored further using the formula \(a^{2}-b^{2}=(a+b)(a-b)\). Thus, it is possible to factor \(4x^{2} - 49\) as \( (2x)^{2} - 7^{2} \), which factors to \( (2x + 7)(2x - 7) \). Consequently, the given expression \(y = 4x^{3} - 49x\) can now be written in factored form as \(y = x(2x + 7)(2x - 7)\)

Step 3: Check Answer by Multiplying Out

To ensure that the function has been factored correctly, expand the factored form again. That would mean multiplying out the terms \(x(2x + 7)(2x - 7)\). Starting with the first two terms, derive \(2x^{2} + 7x\). Multiply this result by \(2x - 7\), and with simplification, get back the original function \(4x^{3} - 49x\), which confirms that the function was factored correctly.

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic tool used to factor expressions. When you have an equation like \[a^2 - b^2\],it can be factored into \[(a + b)(a - b)\].This formula is applicable because it reveals the symmetry between the sum and the difference of two terms. You can identify the difference of squares readily by spotting squares in the terms of the polynomial that are separated by a subtraction sign.
For example, consider the expression \(4x^2 - 49\).Here, \(4x^2\)is equivalent to \((2x)^2\),and \(49\)is equivalent to \(7^2\).This means the expression fits the difference of squares format.
Thus, we can apply the formula and factor it as \((2x + 7)(2x - 7)\).Understanding this formula helps in simplifying many polynomial expressions effectively, especially those that recognize this pattern.

Common Factor

A common factor in algebra is a term that can be uniformly divided out from a polynomial's expression without leaving any remainders. When factoring polynomials, spotting and extracting the common factor is often the first step.
Take the example of \(4x^3 - 49x\).Both terms of this expression share a common factor of \(x\),which can be factored out to simplify the expression before applying any other factoring technique.
When you factor out \(x\),you rewrite the expression as \(x(4x^2 - 49)\).This extraction not only simplifies the process of further factoring but also highlights other properties of the polynomial, as seen here after removing \(x\).
Identifying and factoring out common factors is key in managing larger and more complex polynomial problems by reducing their degrees.

Multiplying Polynomials

Multiplying polynomials is both a useful skill and a method for checking your factoring work. It involves distributing each term of one polynomial to every term of another. This process is known as the distributive property or the FOIL method when dealing with binomials.
In the exercise, we factored the polynomial expression to \(x(2x + 7)(2x - 7)\).To verify this factorization, you multiply the factors back together.
Start by multiplying \((2x + 7)\)and \((2x - 7)\).Using the distributive property, this expands to \(4x^2 - 49\).Then, multiply this result by \(x\),returning to the original polynomial \(4x^3 - 49x\).This reverse process confirms not only your factoring but also assures no steps were missed, guaranteeing the accuracy of the work.