Perform the indicated operation or operations. (3x+5)(2x−9)−(7x−2)(x−1)

Verified step by step guidance

1. First, apply the distributive property (also known as the FOIL method for binomials) to expand each product separately. For the first product, expand \((3x+5)(2x-9)\) by multiplying each term in the first binomial by each term in the second binomial.
2. Next, expand the second product \((7x-2)(x-1)\) in the same way, multiplying each term in the first binomial by each term in the second binomial.
3. After expanding both products, write down the full expression with the expanded terms: the result of the first product minus the result of the second product.
4. Combine like terms by grouping the terms with \(x^2\), the terms with \(x\), and the constant terms separately. This will simplify the expression into a single polynomial.
5. Finally, write the simplified polynomial expression as your answer, ensuring all like terms are combined and the expression is in standard form (descending powers of \(x\)).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves applying the distributive property to multiply each term in one polynomial by every term in the other. For binomials, this often uses the FOIL method (First, Outer, Inner, Last) to systematically multiply terms and combine like terms.

Combining Like Terms

After expanding polynomials, like terms—terms with the same variable raised to the same power—must be combined by adding or subtracting their coefficients. This simplifies the expression into a standard polynomial form.

Subtraction of Polynomials

Subtracting polynomials requires distributing the negative sign across all terms of the polynomial being subtracted before combining like terms. This ensures correct sign changes and accurate simplification of the expression.