Find the real zeros of the function algebraically

[swatmedic05]

f(x)=x^3+5x^2+9x+45
x^3+5x^2+9x+45=0
x^2(x+5)+9(x-5)=0
(x^2+9)(x+5)=0
What happends to the (x+5)?

 

[Mark44]

f(x)=x^3+5x^2+9x+45
x^3+5x^2+9x+45=0
x^2(x+5)+9(x-5)=0

There's an error in the step above.

(x^2+9)(x+5)=0
What happends to the (x+5)?

Since (x² + 9)(x + 5) = 0, then either x² + 9 = 0 or x + 5 = 0. Can you continue from here?

 

[eumyang]

$$\begin{aligned}<br /> x^3 + 5x^2 + 9x + 45 &= 0 \\<br /> x^2(x + 5) + 9(x + 5) &= 0 \\<br /> (x^2 + 9)(x + 5) &= 0<br /> \end{aligned}$$
I'm assuming that the error pointed out above is just a typo, so I'll just fix it. As for

What happened to the (x+5)?

It was the greatest common factor, so it got factored out.

Say you wanted to factor 16x - 28. The GCF is 4, and you could rewrite the expression as
4(4x) - 4(7),
and then factor out the 4 to get
4(4x - 7).

It's the same thing in your problem. The GCF is an expression: x + 5. You factor the x + 5 out in $$x^2 \bold{(x + 5)} + 9 \bold{(x + 5)} = 0$$ and you get $$(x^2 + 9)(x + 5) = 0$$. Maybe part of the confusion is that the GCF appears last instead of first?69

 

[HallsofIvy]

$$\begin{aligned}<br /> x^3 + 5x^2 + 9x + 45 &= 0 \\<br /> x^2(x + 5) + 9(x + 5) &= 0 \\<br /> (x^2 + 9)(x + 5) &= 0<br /> \end{aligned}$$
I'm assuming that the error pointed out above is just a typo, so I'll just fix it. As for

It was the greatest common monomial factor, so it got factored out.

It was the greatest linear factor. It is not a monomial, it is a binomial, just like $x^2+ 9$.

Say you wanted to factor 16x - 28. The GCF is 4, and you could rewrite the expression as
4(4x) - 4(7),
and then factor out the 4 to get
4(4x - 7).

It's the same thing in your problem. The GCF is an expression: x + 5. You factor the x + 5 out in $$x^2 \bold{(x + 5)} + 9 \bold{(x + 5)} = 0$$ and you get $$(x^2 + 9)(x + 5) = 0$$. Maybe part of the confusion is that the GCF appears last instead of first?


69

 

[eumyang]

Fixed. Didn't mean to use the term "monomial" -- just GCF. This is what you get for replying at 5am. :P69