Distributing a Negative Monomial Over a Polynomial
Distributing a negative monomial over a polynomial is a Grade 9 Algebra 1 skill in California Reveal Math (Unit 9: Polynomials). The distributive property requires multiplying the negative monomial by every term in the polynomial: -a(b + c) = -ab - ac. For -3x(2x^2 + 5x - 4): multiply each term to get -6x^3 - 15x^2 + 12x. A negative times a negative term produces a positive. The most common error is failing to distribute the negative to every term.
Key Concepts
When multiplying a polynomial by a negative monomial, apply the distributive property to every term in the polynomial, including the negative sign:.
$$ a(b + c) = ab + ( ac) = ab ac$$.
Common Questions
How do you distribute -3x over (2x^2 + 5x - 4)?
-3x * 2x^2 = -6x^3, then -3x * 5x = -15x^2, then -3x * (-4) = +12x. Result: -6x^3 - 15x^2 + 12x. The negative monomial times a negative term gives a positive product.
How do you distribute -2x^2 over (4x^3 - 7x + 1)?
-2x^2 * 4x^3 = -8x^5, then -2x^2 * (-7x) = +14x^3, then -2x^2 * 1 = -2x^2. Result: -8x^5 + 14x^3 - 2x^2.
What is the most common error when distributing a negative monomial?
Forgetting to distribute the negative sign to every term. For example, writing -3x(2x^2 + 5x - 4) = -6x^3 - 15x^2 - 12x instead of +12x for the last term.
What rule determines the sign of each product?
Negative times positive = negative. Negative times negative = positive. Apply standard sign rules when the negative monomial multiplies each term, tracking whether each polynomial term is positive or negative.
How do you find the exponent of each product term?
Add the exponents of like bases. For -3x * 5x: coefficients give -15, and x * x = x^2. For -3x * 2x^2: coefficients give -6, and x * x^2 = x^3.