High School Number Sense Lesson 33: Substitution (Direct)
Direct substitution questions appeared more times than there were tests this year: 22 times, on average at question # 33. They look different from functions but operate in the same way.
Number Dojo Level: 139
For these question types, you will be given one or more variables and their equivalents, and a polynomial. You will be asked to find the value of the polynomial. A simple version is this:
For many of these questions, it will save you lots of time if you know how to factor polynomials. To learn how to do this, see https://www.khanacademy.org/math/algebra/polynomial-factorization.
Example 1: Let x = -2. Find 3 - 4x. ___
Example 2: Let x = -4. Find 3x - 2. ___
Example 3:
Direct substitution questions appeared more times than there were tests this year: 22 times, on average at question # 33. They look different from functions but operate in the same way.
Number Dojo Level: 139
For these question types, you will be given one or more variables and their equivalents, and a polynomial. You will be asked to find the value of the polynomial. A simple version is this:
- Let x = -5. Find 4 + 3x. ___
- 4 + 3x = 4 + 3(-5) = 4 + (-15) = -11
For many of these questions, it will save you lots of time if you know how to factor polynomials. To learn how to do this, see https://www.khanacademy.org/math/algebra/polynomial-factorization.
Example 1: Let x = -2. Find 3 - 4x. ___
- 3 - 4x = 3 - 4(-2) = 3 + 8 = 11
Example 2: Let x = -4. Find 3x - 2. ___
- 3x - 2 = 3(-4) - 2 = -12 - 2 = -14
Example 3:
If x = 2 and y = 5, then 4x2 + 4xy + y2 = ___
1. This polynomial can be factored to become (2x + y)2.
2. Substitute 2 for x and 5 for y: (2x + y)2 = [2(2) + 5]2 = (4 + 5)2 = 92 = 81.
1. This polynomial can be factored to become (2x + y)2.
2. Substitute 2 for x and 5 for y: (2x + y)2 = [2(2) + 5]2 = (4 + 5)2 = 92 = 81.
Example 4:
If x = 5 and y = -1, then x2 + 2xy + y2 = ___
1. This polynomial can be factored to become (x + y)2.
2. Substitute 5 for x and -1 for y: (x + y)2 = [5 + (-1)]2 = (5 - 1)2 = 42 = 16.
1. This polynomial can be factored to become (x + y)2.
2. Substitute 5 for x and -1 for y: (x + y)2 = [5 + (-1)]2 = (5 - 1)2 = 42 = 16.
Example 5:
If x = 5 and y = 3, then 9x2 - 6xy + y2 = ___
1. This polynomial can be factored to become (3x - y)2.
2. Substitute 5 for x and 3 for y: (3x - y)2 = [3(5) - 3]2 = (15 - 3)2 = 122 = 144.
1. This polynomial can be factored to become (3x - y)2.
2. Substitute 5 for x and 3 for y: (3x - y)2 = [3(5) - 3]2 = (15 - 3)2 = 122 = 144.
Example 6:
If x = 5 and y = 2, then (x - y)(x2 + 2xy + y2) equals ___
1. As seen above, (x2 + 2xy + y2) can be factored as (x + y)2. So now we have (x - y)(x + y)2.
2. Substitute 5 for x and 2 for y: (x - y)(x + y)2 = (5 - 2)(5 + 2)2 = (3)(7)2 = 3(49) = 147.
1. As seen above, (x2 + 2xy + y2) can be factored as (x + y)2. So now we have (x - y)(x + y)2.
2. Substitute 5 for x and 2 for y: (x - y)(x + y)2 = (5 - 2)(5 + 2)2 = (3)(7)2 = 3(49) = 147.
Here's a free worksheet to help you practice SubstDirect:
| substdirect.pdf |
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