Multiplying Functions

Front

What is the product of two functions?

Back

The product of two functions, denoted as \( f(x) \cdot g(x) \), is a new function formed by multiplying the outputs of the two functions for each input value of \( x \).

Front

If \( m(x) = 2x - 1 \) and \( n(x) = -x^3 - 1 \), what is \( m(x) \cdot n(x) \)?

Back

\( m(x) \cdot n(x) = (2x - 1)(-x^3 - 1) = -2x^4 + x^3 + 2x - 1 \) (correct answer: \( -2x^4 + x^3 - 2x + 1 \)).

Front

If \( d(x) = 4x + 4 \) and \( c(x) = 2x - 4 \), what is \( d(x) \cdot c(x) \)?

Back

\( d(x) \cdot c(x) = (4x + 4)(2x - 4) = 8x^2 - 8x - 16 \).

Front

What is the formula for multiplying two binomials?

Back

The formula for multiplying two binomials \( (a + b)(c + d) \) is given by: \( ac + ad + bc + bd \).

Front

If \( f(x) = 3x - 1 \) and \( g(x) = x^3 + 5x^2 \), what is \( f(x) \cdot g(x) \)?

Back

\( f(x) \cdot g(x) = (3x - 1)(x^3 + 5x^2) = 3x^4 + 15x^3 - x^3 - 5x^2 = 3x^4 + 14x^3 - 5x^2 \).

Front

What is the degree of the product of two polynomials?

Back

The degree of the product of two polynomials is the sum of their degrees. If \( f(x) \) has degree \( m \) and \( g(x) \) has degree \( n \), then \( \text{deg}(f(x) \cdot g(x)) = m + n \).

Front

If \( A(x) = 4x - 2 \) and \( B(x) = n^2 + 1 \), what is \( A(x) \cdot B(x) \)?

Back

\( A(x) \cdot B(x) = (4x - 2)(n^2 + 1) = 4xn^2 + 4x - 2n^2 - 2 \).

Front

What is the zero product property?

Back

The zero product property states that if \( a \cdot b = 0 \), then either \( a = 0 \) or \( b = 0 \).

Front

If \( A(x) = -x - 2 \) and \( B(x) = 4x \), what is \( A(x) \cdot B(x) \)?

Back

\( A(x) \cdot B(x) = (-x - 2)(4x) = -4x^2 - 8x \).

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It determines the end behavior of the polynomial function.