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Answer :
Sure, let's examine each expression to see if they can be rewritten in a quadratic form.
In order to write an expression in quadratic form, we generally look for a pattern like [tex]\(a \cdot x^{2n} + b \cdot x^n + c\)[/tex], where the degree of each term follows this sequence: [tex]\(2n, n, 0\)[/tex].
Let's analyze each expression:
16. [tex]\(x^4 + 12x^2 - 8\)[/tex]:
- The degrees of the terms are 4, 2, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 2\)[/tex].
- Thus, this expression can be written in quadratic form.
17. [tex]\(-15x^4 + 18x^2 - 4\)[/tex]:
- The degrees of the terms are 4, 2, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 2\)[/tex].
- Thus, this expression can be written in quadratic form.
18. [tex]\(8x^6 + 6x^3 + 7\)[/tex]:
- The degrees of the terms are 6, 3, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 3\)[/tex].
- Thus, this expression can be written in quadratic form.
19. [tex]\(5x^6 - 2x^2 + 8\)[/tex]:
- The degrees of the terms are 6, 2, and 0.
- This pattern does not fit the requirement for forming a quadratic pattern because one of the terms (specifically [tex]\(x^2\)[/tex]) doesn't match [tex]\(Ax^n\)[/tex] where [tex]\(n = 3\)[/tex].
- Therefore, this expression cannot be written in quadratic form.
20. [tex]\(9x^8 - 21x^4 + 12\)[/tex]:
- The degrees of the terms are 8, 4, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 4\)[/tex].
- Thus, this expression can be written in quadratic form.
21. [tex]\(16x^{10} + 2x^5 + 6\)[/tex]:
- The degrees of the terms are 10, 5, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 5\)[/tex].
- Thus, this expression can be written in quadratic form.
So, the expressions that can be rewritten in quadratic form are: 16, 17, 18, 20, and 21. The expression 19 cannot be rewritten in quadratic form.
In order to write an expression in quadratic form, we generally look for a pattern like [tex]\(a \cdot x^{2n} + b \cdot x^n + c\)[/tex], where the degree of each term follows this sequence: [tex]\(2n, n, 0\)[/tex].
Let's analyze each expression:
16. [tex]\(x^4 + 12x^2 - 8\)[/tex]:
- The degrees of the terms are 4, 2, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 2\)[/tex].
- Thus, this expression can be written in quadratic form.
17. [tex]\(-15x^4 + 18x^2 - 4\)[/tex]:
- The degrees of the terms are 4, 2, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 2\)[/tex].
- Thus, this expression can be written in quadratic form.
18. [tex]\(8x^6 + 6x^3 + 7\)[/tex]:
- The degrees of the terms are 6, 3, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 3\)[/tex].
- Thus, this expression can be written in quadratic form.
19. [tex]\(5x^6 - 2x^2 + 8\)[/tex]:
- The degrees of the terms are 6, 2, and 0.
- This pattern does not fit the requirement for forming a quadratic pattern because one of the terms (specifically [tex]\(x^2\)[/tex]) doesn't match [tex]\(Ax^n\)[/tex] where [tex]\(n = 3\)[/tex].
- Therefore, this expression cannot be written in quadratic form.
20. [tex]\(9x^8 - 21x^4 + 12\)[/tex]:
- The degrees of the terms are 8, 4, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 4\)[/tex].
- Thus, this expression can be written in quadratic form.
21. [tex]\(16x^{10} + 2x^5 + 6\)[/tex]:
- The degrees of the terms are 10, 5, and 0.
- This follows the pattern [tex]\(x^{2n} + Ax^n + B\)[/tex] with [tex]\(n = 5\)[/tex].
- Thus, this expression can be written in quadratic form.
So, the expressions that can be rewritten in quadratic form are: 16, 17, 18, 20, and 21. The expression 19 cannot be rewritten in quadratic form.
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