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| 10 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Consider the following algebraic expressions. (n+2)^2-4 and n^2+4n These algebraic expressions are equivalent.
There can be different methods for factoring a quadratic expression, depending on its type. If the expression is the square of a binomial, it can be factored as a perfect square trinomial.
A perfect square trinomial is a trinomial that can be written as the square of a binomial. There are two general types of perfect square trinomials that can be useful when dealing with quadratic functions or quadratic equations.
Perfect Square Trinomial | Square of Binomial |
---|---|
a^2+2 a b+ b^2 | ( a+ b)^2 |
a^2-2 a b+ b^2 | ( a- b)^2 |
One good way to recognize if a trinomial is a perfect square trinomial is to look at its first and last terms. If they are both perfect squares, there is a good chance that it is a perfect square trinomial. In the given expression, the first and last terms can be written as the squares of 4x and 11, respectively. 16x^2+88x+121 ⇓ ( 4x)^2+88x+( 11)^2 These perfect squares show that the expression could be a perfect square trinomial. However, this is not enough to decide yet.
The next step is to check whether the middle term is two times the square roots of the first and last terms. ( 4x)^2+88x+( 11)^2 ⇓ ( 4x)^2+2( 4x)( 11)+( 11)^2 It can be seen that the given expression satisfies this condition as well.
Since the expression satisfies both conditions, it is a perfect square trinomial. Therefore, it can be written as a square of a binomial where 4x and 11 are the first and second terms of the binomial, respectively. 16x^2+88x+121=( 4x+ 11)^2
Jordan has two copies of the same square painting, and she wants to frame these copies. The area of each painting is represented by a quadratic expression.
Begin by determining whether the expressions are perfect square trinomials.
The side length of a square can be found by taking the square root of its area. Side Length = sqrt(A) If the given expressions are perfect square trinomials, each can be factored and written as a square of a binomial. Therefore, it first needs to be determined whether the given expressions are perfect square trinomials. To do so, begin by checking whether the first and last terms of the expressions are perfect squares.
Expression | First Term | Last Term |
---|---|---|
49x^2-56x+16 | 49x^2=( 7x)^2 | 16=( 4)^2 |
16x^2+24x+9 | 16x^2=( 4x)^2 | 9=( 3)^2 |
Since the first and last terms are perfect squares, the trinomials are probably perfect square trinomials. To be sure, check whether the middle terms are two times the square roots of the first and last terms.
Expression | First Term | Last Term | Middle Term |
---|---|---|---|
49x^2-56x+16 | 49x^2=( 7x)^2 | 16=( 4)^2 | 56x=2* 7x* 4 |
16x^2+24x+9 | 16x^2=( 4x)^2 | 9=( 3)^2 | 24x=2* 4x* 3 |
The expressions satisfy the conditions to be a perfect square trinomials. From here, each trinomial can be written as a square of a binomial. 49x^2-56x+16 & = ( 7x- 4)^2 16x^2+24x+9 & = ( 4x + 3)^2 By taking the square roots of these expressions, the side lengths of the paintings can be found. Side Length of Larger Painting sqrt(( 7x- 4)^2) = 7x- 4 [1em] Side Length of Smaller Painting sqrt(( 4x + 3)^2) = 4x + 3
See solution.
Determining if the trinomial is a perfect square trinomial, then factor it.
Looking at the given answers, both Ali and Jordan consider that the given expression is a perfect square trinomial because both find the square of a binomial.
Therefore, it would be a good idea to start by checking if the given expression is indeed a perfect square trinomial. To do this, check if the first and last terms of the expression are perfect squares. 121m^2l ^2-44ml c+4c^2 ⇓ ( 11ml )^2-44ml c+( 2c)^2 As it can be seen, the first and last terms are perfect squares. This means that the given expression could be a perfect square trinomial. To be sure, check whether the middle term is two times the square roots of the first and last terms. ( 11ml )^2-44ml c+( 2c)^2 ⇓ ( 11ml )^2-2( 11ml )( 2c)+( 2c)^2 Since the expression satisfies the conditions, it is a perfect square trinomial and can be written as a square of a binomial. 121m^2l ^2-44ml c+4c^2 ⇓ ( 11ml )^2-2( 11ml )( 2c)+( 2c)^2 ⇓ ( 11ml - 2c)^2 With this information, it can be concluded that Jordan is correct. On the other hand, Ali only took the square root of the coefficients of the first and last terms, so he did not get the correct answer.
Note that the product of two conjugate binomials results in a difference of two squares. Therefore, the difference of two squares can be factored out using reverse thinking.
To factor an expression as a difference of two squares, the terms of the expression should be perfect squares. 9x^2-121 ⇓ ( 3x)^2-( 11)^2 As it can be seen, the terms of the expression are perfect squares.
The volume of a rectangular prism is given by the expression 81x^4-16.
(9x^2+4), (3x+2), and (3x-2)
Factor the expression that represents the volume of the prism.
Split into factors
a* a=a^2
a^m* b^m=(a * b)^m
Split into factors
a* a=a^2
a^m* b^m=(a * b)^m
Tadeo's older brother is in college. He currently lives in a one-room square apartment, from which a small square is cut out as shown in the diagram.
He wants to move to a rectangular room with the same area as his current room.
Therefore, the area of the larger square and the area of the smaller square need to be found first. Recall that area of a square is the square of its side length.
Square | Side Length, s | Area, s^2 |
---|---|---|
Larger Square | 4n+1 | ( 4n+1)^2 |
Smaller Square | 5 | 5^2 |
a^2-b^2=(a+b)(a-b)
Add and subtract terms
Two square windows of a house and their areas are shown in the given image.
Write an expression that represents the difference between the areas of the windows. Show two different ways to find the solution.
& 25x^2-30x+9 - & x^2-14x+49 & 24x^2-16x-40 The resulting expression represents the difference of the areas.
Expression | First Term | Last Term |
---|---|---|
25x^2-30x+9 | 25x^2=( 5x)^2 | 9=( 3)^2 |
x^2-14x+49 | x^2=( x)^2 | 49=( 7)^2 |
Since the first and last terms are perfect squares, there is a good chance that these expressions are perfect square trinomials. Now, check whether the middle terms are two times the square roots of the first and last terms.
Expression | First Term | Last Term | Middle Term |
---|---|---|---|
25x^2-30x+9 | 25x^2=( 5x)^2 | 9=( 3)^2 | 30x=2* 5x * 3 |
x^2-14x+49 | x^2=( x)^2 | 49=( 7)^2 | 14x=2* x * 7 |
a^2-b^2=(a+b)(a-b)
Distribute -1
Add and subtract terms
Throughout the lesson, two methods have been covered for factoring special products. These methods together can be used to solve the challenge presented at the beginning of the lesson. Recall that there were two algebraic expressions. n^2+4n and (n+2)^2-4 These expressions are known to be equivalent.
The number of dots in each figure in terms of n can be obtained in two different ways, one represented by n^2+4n and the other represented by (n+2)^2−4. To illustrate n^2+4n, assume that n^2 is the inside full square of dots and 4n is the four outside borders with n dots each.
Notice that the number of dots in the outside border and side length of the inside square, in terms of n, match the figure number. Now, to illustrate (n+2)^2-4, imagine the larger square with the four additional dots filled in at the corners. Then, (n+2)^2 would be the number of dots in the larger square, since the missing 4 dots were added.
As it can be seen, the number of dots in n^(th) figure can be represented by n^2+4n and by (n+2)^2-4. Therefore, these expressions are equivalent.
Write as a power
a^2-b^2=(a+b)(a-b)
Add and subtract terms
Distribute n
Multiply
Factor each polynomial by using the perfect square trinomial pattern and the difference of squares.
We need to rewrite the given sum as a product to factor it. Let's start by rewriting the first term by using the Product of Powers Property.
After factoring our x^a from the expression, we ended up with a trinomial inside the parentheses. If we factorize the middle term 10x and write the last term 25 as a power, the trinomial can be further factorized by using the perfect square trinomial pattern.
Once again, we will start by looking for a common factor. Since a^b is a common factor, we will first factor it out.
We will then write the expression inside the parentheses as a difference of two squares to factor it completely.