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Answer :
To solve this problem, we need to determine which polynomial has a graph with even symmetry. A polynomial has even symmetry if all the exponents of the variable [tex]\(x\)[/tex] are even.
Let's go through each option:
a) [tex]\(8x^3 + 12x + 6\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 3 (from [tex]\(8x^3\)[/tex]) and 1 (from [tex]\(12x\)[/tex]). Both are odd, so this polynomial does not have even symmetry.
b) [tex]\(7x^4 + 9x^2 - 12\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 4 (from [tex]\(7x^4\)[/tex]) and 2 (from [tex]\(9x^2\)[/tex]). Both are even. Additionally, the constant term -12 is equivalent to [tex]\(12x^0\)[/tex] where the exponent 0 is considered even. Therefore, this polynomial has even symmetry.
c) [tex]\(9x^7 - 4x^6 + 18x^2 - 7\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 7, 6, and 2. The exponent 7 is odd, so this polynomial does not have even symmetry.
d) [tex]\(8x^6 - 6x^4 + 4x\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 6 (from [tex]\(8x^6\)[/tex]), 4 (from [tex]\(6x^4\)[/tex]), and 1 (from [tex]\(4x\)[/tex]). The exponent 1 is odd, so this polynomial does not have even symmetry.
From this analysis, the polynomial [tex]\(7x^4 + 9x^2 - 12\)[/tex] is the one that has even symmetry. Therefore, the answer is:
Polynomial b) [tex]\(7x^4 + 9x^2 - 12\)[/tex] has a graph with even symmetry.
Let's go through each option:
a) [tex]\(8x^3 + 12x + 6\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 3 (from [tex]\(8x^3\)[/tex]) and 1 (from [tex]\(12x\)[/tex]). Both are odd, so this polynomial does not have even symmetry.
b) [tex]\(7x^4 + 9x^2 - 12\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 4 (from [tex]\(7x^4\)[/tex]) and 2 (from [tex]\(9x^2\)[/tex]). Both are even. Additionally, the constant term -12 is equivalent to [tex]\(12x^0\)[/tex] where the exponent 0 is considered even. Therefore, this polynomial has even symmetry.
c) [tex]\(9x^7 - 4x^6 + 18x^2 - 7\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 7, 6, and 2. The exponent 7 is odd, so this polynomial does not have even symmetry.
d) [tex]\(8x^6 - 6x^4 + 4x\)[/tex]
- The exponents of [tex]\(x\)[/tex] are 6 (from [tex]\(8x^6\)[/tex]), 4 (from [tex]\(6x^4\)[/tex]), and 1 (from [tex]\(4x\)[/tex]). The exponent 1 is odd, so this polynomial does not have even symmetry.
From this analysis, the polynomial [tex]\(7x^4 + 9x^2 - 12\)[/tex] is the one that has even symmetry. Therefore, the answer is:
Polynomial b) [tex]\(7x^4 + 9x^2 - 12\)[/tex] has a graph with even symmetry.
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