Factor by any method. See Examples 1–7. $4z^4-7z^2-15$

Verified step by step guidance

1. Recognize that the expression \$4z^4 - 7z^2 - 15\( is a quadratic in form, where the variable is \)z^2\(. To make it clearer, let \)u = z^2\(, so the expression becomes \)4u^2 - 7u - 15$.
2. Next, factor the quadratic expression \$4u^2 - 7u - 15\(. To do this, look for two numbers that multiply to \(4 \times (-15) = -60\) and add up to \)-7$.
3. Once you find those two numbers, rewrite the middle term \(-7u\) as the sum of two terms using these numbers. This will allow you to factor by grouping.
4. Group the terms into two pairs and factor out the greatest common factor (GCF) from each group.
5. Finally, factor out the common binomial factor from the two groups, and then substitute back \(z^2\) for \(u\) to write the fully factored expression in terms of \(z\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process helps simplify expressions and solve equations. Common methods include factoring out the greatest common factor, grouping, and using special formulas like difference of squares or quadratic forms.

Substitution Method for Quadratic Form

When a polynomial has terms with powers that are multiples of a common base (e.g., z^4 and z^2), substitution can simplify it into a quadratic form. For example, letting u = z^2 transforms 4z^4 - 7z^2 - 15 into 4u^2 - 7u - 15, which can be factored using quadratic techniques.

Factoring Quadratic Expressions

Factoring quadratic expressions involves finding two binomials whose product equals the quadratic. This often requires identifying two numbers that multiply to the constant term and add to the middle coefficient. Techniques include trial and error, factoring by grouping, or using the quadratic formula to find roots.