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| 10 Theory slides |
| 9 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.
When a binomial is squared, the resulting expression is a perfect square trinomial.
(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2
For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.
(a±b)2=a2±2ab+b2
This rule will be first proven for (a+b)2 and then for (a−b)2.
a2=a⋅a
Distribute (a+b)
Distribute a
Distribute b
Commutative Property of Multiplication
Add terms
a2=a⋅a
Distribute (a−b)
Distribute a
Distribute -b
Commutative Property of Multiplication
Subtract terms
Calculate the area of a square with side lengths x+2 feet. Then, calculate the area of a square with side lengths x−1 feet. Finally, find the difference between these areas.
(a+b)2=a2+2ab+b2
Commutative Property of Multiplication
Calculate power
Multiply
The area of a square is calculated by squaring its side length.
(a+b)2=a2+2ab+b2
(ab)m=ambm
(am)n=am⋅n
Multiply
The product of two conjugate binomials is the difference of two squares.
(a+b)(a−b)=a2−b2
Distribute (a−b)
Distribute a
Distribute b
Commutative Property of Multiplication
Add terms
The area of a square is calculated by squaring its side length. The area of a rectangle is calculated by multiplying the length by the width.
(a+b)2=a2+2ab+b2
(ab)m=ambm
(am)n=am⋅n
Multiply
am⋅an=am+n
(a−b)2=a2−2ab+b2
(am)n=am⋅n
Commutative Property of Multiplication
Multiply
a⋅am=a1+m
(ab)m=ambm
Identity Property of Multiplication
(a+b)(a−b)=a2−b2
(ab)m=ambm
(am)n=am⋅n
1a=1
State the degree and the leading coefficient of the resulting polynomial after squaring the binomial or multiplying the conjugate binomials.
LHS⋅(a+b)=RHS⋅(a+b)
Commutative Property of Multiplication
a⋅am=a1+m
Substitute expressions
a(-b)=-a⋅b
(-a)3=-a3
a+(-b)=a−b
(-a)2=a2
A perfect square trinomial is an expression that is the result of squaring either a sum or a difference of two monomials. ( a± b)^2 &= a^2± 2 a b+ b^2_() & Perfect square trinomial Now, we will consider the given trinomial. 81x^2 - 198x + c Since the sign of the linear term is negative, we are looking for the square of a difference of monomials. To find the value of c, we will first find the values of a and b by using the Properties of Exponents. Let's start by rewriting 81x^2 to find a.
We have found that a= 9x. Now, we will rewrite the middle term 198x to find the value of b. Remember that we are looking for 2 a b. Therefore, we will split 198x into factors in such a way including 9x as a factor in it. 198x ⇒ 2( 9x)( 11) This means that the value of b is 11. Let's substitute the values we know into the formula for the square of a binomial with a negative sign. ( a- b)^2 & = a^2-2 a b+ b^2 [0.5em] & = 9x^2-2( 9x)( 11)+c We can deduce that, in order for the given trinomial to be a perfect square trinomial, the value of c must be 11^2= 121. Since we have completed the given task we can stop here, but let's substitute c=121 into our expression and express the trinomial as the square of a binomial.