Find the product of $(4 z^2+7 z-8)$ and $(-z+3)$. The product of $(4 z^2+7 z-8)$ and $(-z+3)$ is $-4 z^3+$ $\square$ $z^2+$ $\square$ $z-24$
Answer (2)
Multiply the two polynomials ( 4 z 2 + 7 z − 8 ) and ( − z + 3 ) .
Expand the expression: − 4 z 3 + 12 z 2 − 7 z 2 + 21 z + 8 z − 24 .
Combine like terms: − 4 z 3 + ( 12 − 7 ) z 2 + ( 21 + 8 ) z − 24 .
Simplify to get the final product: − 4 z 3 + 5 z 2 + 29 z − 24 .
Explanation
Understanding the Problem We are given two polynomials, 4 z 2 + 7 z − 8 and − z + 3 . Our goal is to find their product and express it in the form − 4 z 3 + □ z 2 + □ z − 24 .
Setting up the Multiplication To find the product, we'll multiply the two polynomials: ( 4 z 2 + 7 z − 8 ) ( − z + 3 ) . We can distribute each term of the first polynomial to each term of the second polynomial.
Expanding the Expression Let's expand the expression:
( 4 z 2 + 7 z − 8 ) ( − z + 3 ) = 4 z 2 ( − z + 3 ) + 7 z ( − z + 3 ) − 8 ( − z + 3 )
= − 4 z 3 + 12 z 2 − 7 z 2 + 21 z + 8 z − 24
Combining Like Terms Now, we combine like terms:
− 4 z 3 + ( 12 z 2 − 7 z 2 ) + ( 21 z + 8 z ) − 24 = − 4 z 3 + 5 z 2 + 29 z − 24
Final Answer The product of the two polynomials is − 4 z 3 + 5 z 2 + 29 z − 24 . Therefore, the missing coefficients are 5 and 29.
Examples
Polynomial multiplication is a fundamental concept in algebra and has numerous real-world applications. For example, engineers use polynomial multiplication to model the behavior of systems, such as the trajectory of a projectile or the growth of a population. In economics, polynomial multiplication can be used to model cost and revenue functions to optimize business decisions. Understanding polynomial multiplication helps in designing efficient algorithms and solving complex problems in various fields.
The product of the polynomials ( 4 z 2 + 7 z − 8 ) and ( − z + 3 ) is − 4 z 3 + 5 z 2 + 29 z − 24 . The missing coefficients are 5 and 29. This process involves distributing and combining like terms from the two expressions.
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