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Perfect square trinomials are a form of quadratic expression that can be factored into the square of a binomial. They have a specific pattern: \\( a^2 + 2ab + b^2 = (a + b)^2 $\)or \\( a^2 - 2ab + b^2 = (a - b)^2 \). Recognizing these patterns is key to factoring them efficiently.
Factoring Perfect Square Trinomials
A perfect square trinomial is formed by squaring a binomial. It looks like this: \\( (x + a)^2 = x^2 + 2ax + a^2 $\)or \\( (x - a)^2 = x^2 - 2ax + a^2 \).
Examples
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Factor \\( x^2 + 6x + 9 \). \\( x^2 + 6x + 9 = (x + 3)^2 \), because \\( 3^2 = 9 $\)and \\( 2 \cdot x \cdot 3 = 6x \).
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Factor \\($ x^2 - 10x + 25 $\). \\( x^2 - 10x + 25 = (x - 5)^2 \), because \\( 5^2 = 25 $\)and \\( -2 \cdot x \cdot 5 = -10x \).
Checking Our Work with R
We can use R to verify the factoring of perfect square trinomials.
# Define the binomial as a function
f1 <- function(x) { (x + 3)^2 }
f2 <- function(x) { (x - 5)^2 }
# Evaluate the function at different points
f1(1) # Should return 16 (1 + 3)^2
f2(2) # Should return 9 (2 - 5)^2
Visualizing Perfect Square Trinomials
We can plot a perfect square trinomial to see its parabolic shape, which is characteristic of quadratic functions.
Plot Example
Let's plot \\($ x^2 + 6x + 9 $\).
x <- seq(-10, 10, by = 0.1)
y <- x^2 + 6*x + 9
plot(x, y, type='l', main="Plot of x^2 + 6x + 9", xlab="x", ylab="y")
Understanding the structure of perfect square trinomials and how to factor them is essential in algebra. It not only simplifies expressions but also aids in solving quadratic equations and understanding the properties of quadratic functions.