Example:
$\checkmark$ Multiply the monomial to each term of the polynomial.
$\underbrace{5 x^2}_{5 x(x+3)}=5 x^2+15 x$
$15 x$
1) $x(x+9)$ $x^2+9 x$
2) $2 x(x+6)$ $2 x^2+12 x$
3) $4\left(x^2+3 x-1\right)$ $8^2+12 x-4$
4) $-3 x(6 x+2)$ $-18 x^2-6 x$
5) $9 x(x+3)^2$ $9 x^2+27 x$
Answer (2)
The problem involves distributing a monomial across each term of a polynomial expression for five given cases. By following the distributive property, we correctly find the resulting polynomial expressions for each case. The answers include x 2 + 9 x , 2 x 2 + 12 x , 4 x 2 + 12 x − 4 , − 18 x 2 − 6 x , and 9 x 3 + 54 x 2 + 81 x .
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Distribute the monomial to each term of the polynomial in the first expression: x ( x + 9 ) = x 2 + 9 x .
Distribute the monomial to each term of the polynomial in the second expression: 2 x ( x + 6 ) = 2 x 2 + 12 x .
Distribute the monomial to each term of the polynomial in the third expression: 4 ( x 2 + 3 x − 1 ) = 4 x 2 + 12 x − 4 .
Distribute the monomial to each term of the polynomial in the fourth expression: − 3 x ( 6 x + 2 ) = − 18 x 2 − 6 x .
Distribute the monomial to each term of the polynomial in the fifth expression: 9 x ( x + 3 ) 2 = 9 x 3 + 54 x 2 + 81 x .
Explanation
Understanding the problem We need to multiply the monomial to each term of the polynomial for the 5 given expressions.
Expression 1
x ( x + 9 ) = x × x + x × 9 = x 2 + 9 x
Expression 2
2 x ( x + 6 ) = 2 x × x + 2 x × 6 = 2 x 2 + 12 x
Expression 3
4 ( x 2 + 3 x − 1 ) = 4 × x 2 + 4 × 3 x + 4 × ( − 1 ) = 4 x 2 + 12 x − 4
Expression 4
− 3 x ( 6 x + 2 ) = − 3 x × 6 x + ( − 3 x ) × 2 = − 18 x 2 − 6 x
Expression 5
9 x ( x + 3 ) 2 = 9 x ( x 2 + 6 x + 9 ) = 9 x × x 2 + 9 x × 6 x + 9 x × 9 = 9 x 3 + 54 x 2 + 81 x
Final Answer The results of multiplying the monomial to each term of the polynomial are:
x 2 + 9 x
2 x 2 + 12 x
4 x 2 + 12 x − 4
− 18 x 2 − 6 x
9 x 3 + 54 x 2 + 81 x
Examples
Understanding how to multiply monomials and polynomials is essential in various fields, such as engineering, physics, and computer graphics. For example, when calculating the area of a rectangular garden where the length is a polynomial expression and the width is a monomial, you need to multiply them to find the total area. Similarly, in physics, when dealing with projectile motion, polynomial expressions often describe the position or velocity of an object, and multiplying these by a monomial (like time) helps predict future states. This skill is also crucial in computer graphics for scaling and transforming objects in 3D space.
