Answer :
To determine if the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] can be written in quadratic form, we'll follow these steps:
1. Identify the Substitution:
Notice that [tex]\(x^6\)[/tex] is the square of [tex]\(x^3\)[/tex]. So, we can introduce the substitution [tex]\(y = x^3\)[/tex]. This means that [tex]\(x^6 = (x^3)^2 = y^2\)[/tex].
2. Rewrite the Expression:
Replace [tex]\(x^6\)[/tex] and [tex]\(x^3\)[/tex] in the expression:
- [tex]\(x^6\)[/tex] becomes [tex]\(y^2\)[/tex]
- [tex]\(x^3\)[/tex] becomes [tex]\(y\)[/tex]
So, the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] becomes:
[tex]\[
8(y^2) + 6y + 7
\][/tex]
3. Verify Quadratic Form:
Now, let's check if this expression can be written as a quadratic in terms of [tex]\(y\)[/tex]:
A quadratic expression is generally in the form [tex]\(ay^2 + by + c\)[/tex].
- Here, [tex]\(a = 8\)[/tex] (coefficient of [tex]\(y^2\)[/tex]),
- [tex]\(b = 6\)[/tex] (coefficient of [tex]\(y\)[/tex]),
- [tex]\(c = 7\)[/tex] (constant term).
4. Conclusion:
The expression [tex]\(8y^2 + 6y + 7\)[/tex] fits perfectly into the quadratic form [tex]\(ay^2 + by + c\)[/tex]. Therefore, it is possible to write the given expression as a quadratic in terms of [tex]\(y\)[/tex].
So, the quadratic form of the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] is [tex]\(8y^2 + 6y + 7\)[/tex].
1. Identify the Substitution:
Notice that [tex]\(x^6\)[/tex] is the square of [tex]\(x^3\)[/tex]. So, we can introduce the substitution [tex]\(y = x^3\)[/tex]. This means that [tex]\(x^6 = (x^3)^2 = y^2\)[/tex].
2. Rewrite the Expression:
Replace [tex]\(x^6\)[/tex] and [tex]\(x^3\)[/tex] in the expression:
- [tex]\(x^6\)[/tex] becomes [tex]\(y^2\)[/tex]
- [tex]\(x^3\)[/tex] becomes [tex]\(y\)[/tex]
So, the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] becomes:
[tex]\[
8(y^2) + 6y + 7
\][/tex]
3. Verify Quadratic Form:
Now, let's check if this expression can be written as a quadratic in terms of [tex]\(y\)[/tex]:
A quadratic expression is generally in the form [tex]\(ay^2 + by + c\)[/tex].
- Here, [tex]\(a = 8\)[/tex] (coefficient of [tex]\(y^2\)[/tex]),
- [tex]\(b = 6\)[/tex] (coefficient of [tex]\(y\)[/tex]),
- [tex]\(c = 7\)[/tex] (constant term).
4. Conclusion:
The expression [tex]\(8y^2 + 6y + 7\)[/tex] fits perfectly into the quadratic form [tex]\(ay^2 + by + c\)[/tex]. Therefore, it is possible to write the given expression as a quadratic in terms of [tex]\(y\)[/tex].
So, the quadratic form of the expression [tex]\(8x^6 + 6x^3 + 7\)[/tex] is [tex]\(8y^2 + 6y + 7\)[/tex].
