Find each product. (t+4)(t+4)(t-4)(t-4)

Verified step by step guidance

1. Recognize that the expression is a product of two binomials squared: \((t+4)(t+4)(t-4)(t-4)\) can be grouped as \([(t+4)(t+4)] \times [(t-4)(t-4)]\).
2. Rewrite each group as a square: \((t+4)^2 \times (t-4)^2\).
3. Recall the formula for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). Apply this to \((t+4)^2\) to get \(t^2 + 2 \times t \times 4 + 4^2\).
4. Similarly, apply the formula to \((t-4)^2\) to get \(t^2 - 2 \times t \times 4 + 4^2\).
5. Multiply the two resulting expressions together: \((t^2 + 8t + 16)(t^2 - 8t + 16)\). This is a product of two trinomials which can be expanded using the distributive property (FOIL method).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomial expressions by applying the distributive property. Each term in one polynomial is multiplied by every term in the other, and like terms are combined to simplify the result.

Difference of Squares

The difference of squares is a special product formula: (a + b)(a - b) = a² - b². Recognizing this pattern helps simplify expressions quickly, especially when multiplying conjugate binomials.

Combining Like Terms

After multiplying polynomials, like terms (terms with the same variable raised to the same power) must be combined by adding or subtracting their coefficients to simplify the expression into standard polynomial form.