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Answer :
Certainly! Let's start by analyzing the expression and writing it in quadratic form if possible.
The given expression is:
[tex]\[ 8x^6 + 6x^3 + 7 \][/tex]
To determine if this can be written in quadratic form, notice that the powers of [tex]\(x\)[/tex] are in multiples of 3. We can make a substitution to help rewrite this expression.
### Step-by-Step Solution:
1. Substitution:
- Let's set [tex]\( y = x^3 \)[/tex]. This transforms the expression in terms of [tex]\( y \)[/tex]:
- Note that [tex]\( x^6 \)[/tex] can be written as [tex]\( (x^3)^2 \)[/tex], which becomes [tex]\( y^2 \)[/tex].
2. Rewrite the Original Expression:
- Now, rewrite the expression using the substitution [tex]\( y = x^3 \)[/tex]:
[tex]\[
8(x^3)^2 + 6(x^3) + 7
\][/tex]
- This becomes:
[tex]\[
8y^2 + 6y + 7
\][/tex]
3. Resulting Quadratic Form:
- The expression is now in a quadratic form:
[tex]\[
8y^2 + 6y + 7
\][/tex]
Thus, the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] can be written in quadratic form as [tex]\(8y^2 + 6y + 7\)[/tex] with the substitution [tex]\(y = x^3\)[/tex]. This shows that it is indeed possible to express the original polynomial as a quadratic in terms of [tex]\(y\)[/tex].
The given expression is:
[tex]\[ 8x^6 + 6x^3 + 7 \][/tex]
To determine if this can be written in quadratic form, notice that the powers of [tex]\(x\)[/tex] are in multiples of 3. We can make a substitution to help rewrite this expression.
### Step-by-Step Solution:
1. Substitution:
- Let's set [tex]\( y = x^3 \)[/tex]. This transforms the expression in terms of [tex]\( y \)[/tex]:
- Note that [tex]\( x^6 \)[/tex] can be written as [tex]\( (x^3)^2 \)[/tex], which becomes [tex]\( y^2 \)[/tex].
2. Rewrite the Original Expression:
- Now, rewrite the expression using the substitution [tex]\( y = x^3 \)[/tex]:
[tex]\[
8(x^3)^2 + 6(x^3) + 7
\][/tex]
- This becomes:
[tex]\[
8y^2 + 6y + 7
\][/tex]
3. Resulting Quadratic Form:
- The expression is now in a quadratic form:
[tex]\[
8y^2 + 6y + 7
\][/tex]
Thus, the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] can be written in quadratic form as [tex]\(8y^2 + 6y + 7\)[/tex] with the substitution [tex]\(y = x^3\)[/tex]. This shows that it is indeed possible to express the original polynomial as a quadratic in terms of [tex]\(y\)[/tex].
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Rewritten by : Barada
