Answer :
To determine which polynomial has a graph with even symmetry, we need to understand what even symmetry means for a polynomial's graph. A graph is symmetric about the y-axis (even symmetry) if for every point [tex]\((x, y)\)[/tex] on the graph, the point [tex]\((-x, y)\)[/tex] is also on the graph.
For a polynomial to have even symmetry, all the terms in the polynomial must only have even exponents of [tex]\(x\)[/tex]. Let's analyze each option one by one:
a) [tex]\(9x^7 - 4x^6 + 18x^2 - 7\)[/tex]
- Exponents: 7 (odd), 6 (even), 2 (even), 0 (even)
- Since there is a term with an odd exponent ([tex]\(x^7\)[/tex]), this polynomial does not have even symmetry.
b) [tex]\(8x^6 - 6x^4 + 4x\)[/tex]
- Exponents: 6 (even), 4 (even), 1 (odd)
- There is a term with an odd exponent ([tex]\(x\)[/tex]), so this polynomial does not have even symmetry.
c) [tex]\(7x^4 + 9x^2 - 12\)[/tex]
- Exponents: 4 (even), 2 (even), 0 (even)
- All exponents are even. This means the polynomial is symmetric about the y-axis, and it has even symmetry.
d) [tex]\(8x^3 + 12x + 6\)[/tex]
- Exponents: 3 (odd), 1 (odd), 0 (even)
- This polynomial includes terms with odd exponents ([tex]\(x^3\)[/tex] and [tex]\(x\)[/tex]), so it does not have even symmetry.
After analyzing each polynomial, we can conclude that the polynomial with even symmetry is:
c) [tex]\(7x^4 + 9x^2 - 12\)[/tex]
For a polynomial to have even symmetry, all the terms in the polynomial must only have even exponents of [tex]\(x\)[/tex]. Let's analyze each option one by one:
a) [tex]\(9x^7 - 4x^6 + 18x^2 - 7\)[/tex]
- Exponents: 7 (odd), 6 (even), 2 (even), 0 (even)
- Since there is a term with an odd exponent ([tex]\(x^7\)[/tex]), this polynomial does not have even symmetry.
b) [tex]\(8x^6 - 6x^4 + 4x\)[/tex]
- Exponents: 6 (even), 4 (even), 1 (odd)
- There is a term with an odd exponent ([tex]\(x\)[/tex]), so this polynomial does not have even symmetry.
c) [tex]\(7x^4 + 9x^2 - 12\)[/tex]
- Exponents: 4 (even), 2 (even), 0 (even)
- All exponents are even. This means the polynomial is symmetric about the y-axis, and it has even symmetry.
d) [tex]\(8x^3 + 12x + 6\)[/tex]
- Exponents: 3 (odd), 1 (odd), 0 (even)
- This polynomial includes terms with odd exponents ([tex]\(x^3\)[/tex] and [tex]\(x\)[/tex]), so it does not have even symmetry.
After analyzing each polynomial, we can conclude that the polynomial with even symmetry is:
c) [tex]\(7x^4 + 9x^2 - 12\)[/tex]
