multiply binomial by trinomial
Check the final answer first, then review the worked steps.
Problem
multiply binomial by trinomial
Answer
\(7x^4 - 4x^3 + 35x^2 - 27x + 4\)
Step-by-step solution
1. Distribute the terms: To find the product of $(7x - 4)(x^3 + 5x - 1)$, we use the distributive property (also known as the FOIL method extended for polynomials). We multiply each term in the first binomial by each term in the trinomial:
$$(7x)(x^3) + (7x)(5x) + (7x)(-1) + (-4)(x^3) + (-4)(5x) + (-4)(-1)$$
2. Perform the multiplications: Calculate the product of each pair of terms:
$$(7x)(x^3) = 7x^4$$
$$(7x)(5x) = 35x^2$$
$$(7x)(-1) = -7x$$
$$(-4)(x^3) = -4x^3$$
$$(-4)(5x) = -20x$$
$$(-4)(-1) = 4$$
3. Combine the results: Write out the full expression:
$$7x^4 + 35x^2 - 7x - 4x^3 - 20x + 4$$
4. Combine like terms and order by degree: Group the terms with the same exponent and simplify:
$$7x^4 - 4x^3 + 35x^2 + (-7x - 20x) + 4$$
$$7x^4 - 4x^3 + 35x^2 - 27x + 4$$