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To find the [tex]$x$[/tex]-intercepts and [tex]$y$[/tex]-intercept of the function [tex]\( g(x) = (x+1)(x^2 - 10x + 24) \)[/tex], let's proceed with a step-by-step solution.
Step 1: Finding the [tex]$x$[/tex]-intercepts
The [tex]$x$[/tex]-intercepts occur where [tex]\( g(x) = 0 \)[/tex]. So, we need to solve the equation:
[tex]\[ (x + 1)(x^2 - 10x + 24) = 0 \][/tex]
This equation is satisfied when either factor is zero:
[tex]\[ x + 1 = 0 \quad \text{or} \quad x^2 - 10x + 24 = 0 \][/tex]
Solving [tex]\( x + 1 = 0 \)[/tex]:
[tex]\[ x = -1 \][/tex]
Now, solving the quadratic equation [tex]\( x^2 - 10x + 24 = 0 \)[/tex], we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 24 \)[/tex]:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{100 - 96}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{4}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm 2}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{10 + 2}{2} = 6 \][/tex]
[tex]\[ x = \frac{10 - 2}{2} = 4 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -1, 4, 6 \)[/tex].
Step 2: Finding the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. We find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = (0+1)(0^2 - 10 \cdot 0 + 24) \][/tex]
[tex]\[ g(0) = 1 \cdot 24 = 24 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( 24 \)[/tex].
Step 3: Conclusion
The [tex]$x$[/tex]-intercepts are [tex]\( -1, 4, 6 \)[/tex] and the [tex]$y$[/tex]-intercept is [tex]\( 24 \)[/tex].
By analyzing the options provided in the question:
A. [tex]$x$[/tex]-intercepts: [tex]$-1, 4, 6$[/tex]; [tex]$y$[/tex]-intercept: 24
B. [tex]$x$[/tex]-intercepts: [tex]$-6, -4$[/tex], and [tex]$1$[/tex]; [tex]$y$[/tex]-intercept: 24
C. [tex]$x$[/tex]-intercepts: [tex]$-4, -1, 6$[/tex]; [tex]$y$[/tex]-intercept: -24
D. [tex]$x$[/tex]-intercepts: [tex]$-6, -1, 4$[/tex]; [tex]$y$[/tex]-intercept: -24
The correct answer is:
A. [tex]$x$[/tex]-intercepts: [tex]$-1, 4, 6$[/tex]; [tex]$y$[/tex]-intercept: 24
Step 1: Finding the [tex]$x$[/tex]-intercepts
The [tex]$x$[/tex]-intercepts occur where [tex]\( g(x) = 0 \)[/tex]. So, we need to solve the equation:
[tex]\[ (x + 1)(x^2 - 10x + 24) = 0 \][/tex]
This equation is satisfied when either factor is zero:
[tex]\[ x + 1 = 0 \quad \text{or} \quad x^2 - 10x + 24 = 0 \][/tex]
Solving [tex]\( x + 1 = 0 \)[/tex]:
[tex]\[ x = -1 \][/tex]
Now, solving the quadratic equation [tex]\( x^2 - 10x + 24 = 0 \)[/tex], we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 24 \)[/tex]:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{100 - 96}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{4}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm 2}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{10 + 2}{2} = 6 \][/tex]
[tex]\[ x = \frac{10 - 2}{2} = 4 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -1, 4, 6 \)[/tex].
Step 2: Finding the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. We find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = (0+1)(0^2 - 10 \cdot 0 + 24) \][/tex]
[tex]\[ g(0) = 1 \cdot 24 = 24 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( 24 \)[/tex].
Step 3: Conclusion
The [tex]$x$[/tex]-intercepts are [tex]\( -1, 4, 6 \)[/tex] and the [tex]$y$[/tex]-intercept is [tex]\( 24 \)[/tex].
By analyzing the options provided in the question:
A. [tex]$x$[/tex]-intercepts: [tex]$-1, 4, 6$[/tex]; [tex]$y$[/tex]-intercept: 24
B. [tex]$x$[/tex]-intercepts: [tex]$-6, -4$[/tex], and [tex]$1$[/tex]; [tex]$y$[/tex]-intercept: 24
C. [tex]$x$[/tex]-intercepts: [tex]$-4, -1, 6$[/tex]; [tex]$y$[/tex]-intercept: -24
D. [tex]$x$[/tex]-intercepts: [tex]$-6, -1, 4$[/tex]; [tex]$y$[/tex]-intercept: -24
The correct answer is:
A. [tex]$x$[/tex]-intercepts: [tex]$-1, 4, 6$[/tex]; [tex]$y$[/tex]-intercept: 24
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