Answer :
Sure! Let's go through the steps to factor the expression completely:
Given expression:
[tex]\[ x^4 - 14x^2 + 45 \][/tex]
### Step 1: Recognize the form
Notice that the expression is a quadratic in terms of [tex]\( x^2 \)[/tex]. Let's set [tex]\( y = x^2 \)[/tex] to simplify our work:
[tex]\[ (x^2)^2 - 14(x^2) + 45 = y^2 - 14y + 45 \][/tex]
### Step 2: Factor the quadratic in [tex]\( y \)[/tex]
Now we need to factor [tex]\( y^2 - 14y + 45 \)[/tex] into two binomials. We do this by finding two numbers that multiply to 45 and add to -14.
The numbers that satisfy this are -5 and -9, because:
[tex]\[ (-5) \cdot (-9) = 45 \][/tex]
[tex]\[ (-5) + (-9) = -14 \][/tex]
So, we can write the quadratic as:
[tex]\[ y^2 - 14y + 45 = (y - 5)(y - 9) \][/tex]
### Step 3: Substitute back [tex]\( y = x^2 \)[/tex]
Recall that [tex]\( y = x^2 \)[/tex]. Substituting [tex]\( y \)[/tex] back, we get:
[tex]\[ (x^2 - 5)(x^2 - 9) \][/tex]
### Step 4: Factor the difference of squares
Next, notice that [tex]\( x^2 - 9 \)[/tex] is a difference of squares. It can be factored further as:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]
Therefore, we can rewrite our factors as:
[tex]\[ (x^2 - 5)(x - 3)(x + 3) \][/tex]
### Final Answer
The completely factored form of the expression [tex]\( x^4 - 14x^2 + 45 \)[/tex] is:
[tex]\[ (x - 3)(x + 3)(x^2 - 5) \][/tex]
And there we have it! The expression is now factored completely.
Given expression:
[tex]\[ x^4 - 14x^2 + 45 \][/tex]
### Step 1: Recognize the form
Notice that the expression is a quadratic in terms of [tex]\( x^2 \)[/tex]. Let's set [tex]\( y = x^2 \)[/tex] to simplify our work:
[tex]\[ (x^2)^2 - 14(x^2) + 45 = y^2 - 14y + 45 \][/tex]
### Step 2: Factor the quadratic in [tex]\( y \)[/tex]
Now we need to factor [tex]\( y^2 - 14y + 45 \)[/tex] into two binomials. We do this by finding two numbers that multiply to 45 and add to -14.
The numbers that satisfy this are -5 and -9, because:
[tex]\[ (-5) \cdot (-9) = 45 \][/tex]
[tex]\[ (-5) + (-9) = -14 \][/tex]
So, we can write the quadratic as:
[tex]\[ y^2 - 14y + 45 = (y - 5)(y - 9) \][/tex]
### Step 3: Substitute back [tex]\( y = x^2 \)[/tex]
Recall that [tex]\( y = x^2 \)[/tex]. Substituting [tex]\( y \)[/tex] back, we get:
[tex]\[ (x^2 - 5)(x^2 - 9) \][/tex]
### Step 4: Factor the difference of squares
Next, notice that [tex]\( x^2 - 9 \)[/tex] is a difference of squares. It can be factored further as:
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]
Therefore, we can rewrite our factors as:
[tex]\[ (x^2 - 5)(x - 3)(x + 3) \][/tex]
### Final Answer
The completely factored form of the expression [tex]\( x^4 - 14x^2 + 45 \)[/tex] is:
[tex]\[ (x - 3)(x + 3)(x^2 - 5) \][/tex]
And there we have it! The expression is now factored completely.
