Factor $8 r^3-64 r^2+r-8$
Answer (1)
Group the terms: ( 8 r 3 − 64 r 2 ) + ( r − 8 ) .
Factor out 8 r 2 from the first group: 8 r 2 ( r − 8 ) + ( r − 8 ) .
Factor out the common factor ( r − 8 ) : ( 8 r 2 + 1 ) ( r − 8 ) .
The factored form of the expression is ( 8 r 2 + 1 ) ( r − 8 ) .
Explanation
Understanding the Expression We are given the expression 8 r 3 − 64 r 2 + r − 8 and asked to analyze it. Our goal is to factor the expression.
Factoring by Grouping We can try factoring by grouping. We group the first two terms and the last two terms: ( 8 r 3 − 64 r 2 ) + ( r − 8 ) . From the first group, we can factor out 8 r 2 , which gives us: 8 r 2 ( r − 8 ) + ( r − 8 ) . Now we can see that ( r − 8 ) is a common factor.
Final Factored Form Factoring out ( r − 8 ) from the entire expression, we get: ( 8 r 2 + 1 ) ( r − 8 ) . This is the factored form of the given expression.
Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in calculus and other higher-level mathematics. For example, if you are designing a bridge, you might need to analyze the stresses and strains on different parts of the structure. These stresses and strains can often be modeled by polynomial equations, and factoring these equations can help you find critical points where the stress is highest. By understanding these critical points, you can ensure that the bridge is strong enough to withstand the forces acting on it.
