Factor the trinomial.
[tex]r^2-13 r+36[/tex]
Answer (2)
Find two numbers that multiply to 36 and add up to -13: -4 and -9.
Express the trinomial in factored form using these numbers: ( r − 4 ) ( r − 9 ) .
Verify the factorization by expanding ( r − 4 ) ( r − 9 ) to obtain r 2 − 13 r + 36 .
The factored form of the trinomial is ( r − 4 ) ( r − 9 ) .
Explanation
Understanding the Problem We are asked to factor the trinomial r 2 − 13 r + 36 . This means we want to find two binomials that, when multiplied together, give us the original trinomial.
Finding the Factors To factor the trinomial r 2 − 13 r + 36 , we need to find two numbers that multiply to 36 (the constant term) and add up to -13 (the coefficient of the r term). Let's list the pairs of factors of 36:
1 and 36 2 and 18 3 and 12 4 and 9 6 and 6
Since we need the factors to add up to -13, we need to consider the negative factors of 36:
-1 and -36 -2 and -18 -3 and -12 -4 and -9 -6 and -6
Now, let's check which pair adds up to -13:
-1 + (-36) = -37 -2 + (-18) = -20 -3 + (-12) = -15 -4 + (-9) = -13 -6 + (-6) = -12
The pair -4 and -9 adds up to -13.
Writing the Factored Form Now that we have found the two numbers, -4 and -9, we can write the factored form of the trinomial as ( r − 4 ) ( r − 9 ) .
Verifying the Factorization To verify our factorization, we can expand ( r − 4 ) ( r − 9 ) :
( r − 4 ) ( r − 9 ) = r ( r − 9 ) − 4 ( r − 9 ) = r 2 − 9 r − 4 r + 36 = r 2 − 13 r + 36
Since the expanded form matches the original trinomial, our factorization is correct.
Final Answer The factored form of the trinomial r 2 − 13 r + 36 is ( r − 4 ) ( r − 9 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and has many real-world applications. For example, suppose you are designing a rectangular garden and you know the area can be represented by the expression r 2 − 13 r + 36 , where r is a variable related to the dimensions of the garden. By factoring this expression into ( r − 4 ) ( r − 9 ) , you determine that the length and width of the garden can be expressed as ( r − 4 ) and ( r − 9 ) . This allows you to plan the layout of your garden based on the possible values of r , ensuring that the area matches your design requirements. Factoring helps in optimizing space and resources in practical design problems.
The trinomial r 2 − 13 r + 36 can be factored into ( r − 4 ) ( r − 9 ) . This is achieved by finding two numbers that multiply to 36 and add to -13, specifically -4 and -9. Verifying the factorization by expanding confirms its accuracy.
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