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To determine which given polynomials are perfect square trinomials, we need to examine each one closely. A perfect square trinomial takes the form:
[tex]\[ ax^2 + bx + c = (dx + e)^2 \][/tex]
which simplifies to:
[tex]\[ a = d^2 \][/tex]
[tex]\[ b = 2de \][/tex]
[tex]\[ c = e^2 \][/tex]
Let's examine each of the provided polynomials:
1. Polynomial: [tex]\(49x^2 - 28x + 16\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 49 \)[/tex]
- [tex]\( b = -28 \)[/tex]
- [tex]\( c = 16 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 49 = d^2 \)[/tex] ⟹ [tex]\( d = 7 \)[/tex]
- [tex]\( 2de = -28 \)[/tex] ⟹ [tex]\( 2(7)e = -28 \)[/tex] ⟹ [tex]\( e = -2 \)[/tex]
- [tex]\( e^2 = 16 \)[/tex] ⟹ [tex]\((-2)^2 = 4\)[/tex], which does not match the given [tex]\( c = 16 \)[/tex]
Thus, this is not a perfect square trinomial.
2. Polynomial: [tex]\(9a^2 - 30a + 25\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 9 \)[/tex]
- [tex]\( b = -30 \)[/tex]
- [tex]\( c = 25 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 9 = d^2 \)[/tex] ⟹ [tex]\( d = 3 \)[/tex]
- [tex]\( 2de = -30 \)[/tex] ⟹ [tex]\( 2(3)e = -30 \)[/tex] ⟹ [tex]\( e = -5 \)[/tex]
- [tex]\( e^2 = 25 \)[/tex] ⟹ [tex]\((-5)^2 = 25 \)[/tex]
This confirms all conditions match, making it a perfect square trinomial:
[tex]\[ 9a^2 - 30a + 25 = (3a - 5)^2 \][/tex]
Thus, this polynomial is a perfect square trinomial.
3. Polynomial: [tex]\(25b^2 - 45b - 81\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 25 \)[/tex]
- [tex]\( b = -45 \)[/tex]
- [tex]\( c = -81 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 25 = d^2 \)[/tex] ⟹ [tex]\( d = 5 \)[/tex]
- [tex]\( 2de = -45 \)[/tex] ⟹ [tex]\( 2(5)e = -45 \)[/tex] ⟹ [tex]\( e = -4.5 \)[/tex]
- [tex]\( e^2 = -81 \)[/tex] ⟹ [tex]\((-4.5)^2 = 20.25\)[/tex], which does not match the given [tex]\( c = -81 \)[/tex]
Thus, this is not a perfect square trinomial.
4. Polynomial: [tex]\(16x^2 - 24x - 9\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 16 \)[/tex]
- [tex]\( b = -24 \)[/tex]
- [tex]\( c = -9 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 16 = d^2 \)[/tex] ⟹ [tex]\( d = 4 \)[/tex]
- [tex]\( 2de = -24 \)[/tex] ⟹ [tex]\( 2(4)e = -24 \)[/tex] ⟹ [tex]\( e = -3 \)[/tex]
- [tex]\( e^2 = -9 \)[/tex] ⟹ [tex]\((-3)^2 = 9\)[/tex], which does not match the given [tex]\( c = -9 \)[/tex]
Thus, this is not a perfect square trinomial.
To summarize, based on our calculations, the perfect square trinomials among the given polynomials are:
- [tex]\( 9a^2 - 30a + 25 \)[/tex]
All other polynomials are not perfect square trinomials.
[tex]\[ ax^2 + bx + c = (dx + e)^2 \][/tex]
which simplifies to:
[tex]\[ a = d^2 \][/tex]
[tex]\[ b = 2de \][/tex]
[tex]\[ c = e^2 \][/tex]
Let's examine each of the provided polynomials:
1. Polynomial: [tex]\(49x^2 - 28x + 16\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 49 \)[/tex]
- [tex]\( b = -28 \)[/tex]
- [tex]\( c = 16 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 49 = d^2 \)[/tex] ⟹ [tex]\( d = 7 \)[/tex]
- [tex]\( 2de = -28 \)[/tex] ⟹ [tex]\( 2(7)e = -28 \)[/tex] ⟹ [tex]\( e = -2 \)[/tex]
- [tex]\( e^2 = 16 \)[/tex] ⟹ [tex]\((-2)^2 = 4\)[/tex], which does not match the given [tex]\( c = 16 \)[/tex]
Thus, this is not a perfect square trinomial.
2. Polynomial: [tex]\(9a^2 - 30a + 25\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 9 \)[/tex]
- [tex]\( b = -30 \)[/tex]
- [tex]\( c = 25 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 9 = d^2 \)[/tex] ⟹ [tex]\( d = 3 \)[/tex]
- [tex]\( 2de = -30 \)[/tex] ⟹ [tex]\( 2(3)e = -30 \)[/tex] ⟹ [tex]\( e = -5 \)[/tex]
- [tex]\( e^2 = 25 \)[/tex] ⟹ [tex]\((-5)^2 = 25 \)[/tex]
This confirms all conditions match, making it a perfect square trinomial:
[tex]\[ 9a^2 - 30a + 25 = (3a - 5)^2 \][/tex]
Thus, this polynomial is a perfect square trinomial.
3. Polynomial: [tex]\(25b^2 - 45b - 81\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 25 \)[/tex]
- [tex]\( b = -45 \)[/tex]
- [tex]\( c = -81 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 25 = d^2 \)[/tex] ⟹ [tex]\( d = 5 \)[/tex]
- [tex]\( 2de = -45 \)[/tex] ⟹ [tex]\( 2(5)e = -45 \)[/tex] ⟹ [tex]\( e = -4.5 \)[/tex]
- [tex]\( e^2 = -81 \)[/tex] ⟹ [tex]\((-4.5)^2 = 20.25\)[/tex], which does not match the given [tex]\( c = -81 \)[/tex]
Thus, this is not a perfect square trinomial.
4. Polynomial: [tex]\(16x^2 - 24x - 9\)[/tex]
Matching it with the form [tex]\( ax^2 + bx + c \)[/tex]:
- [tex]\( a = 16 \)[/tex]
- [tex]\( b = -24 \)[/tex]
- [tex]\( c = -9 \)[/tex]
For it to be a perfect square trinomial:
- [tex]\( 16 = d^2 \)[/tex] ⟹ [tex]\( d = 4 \)[/tex]
- [tex]\( 2de = -24 \)[/tex] ⟹ [tex]\( 2(4)e = -24 \)[/tex] ⟹ [tex]\( e = -3 \)[/tex]
- [tex]\( e^2 = -9 \)[/tex] ⟹ [tex]\((-3)^2 = 9\)[/tex], which does not match the given [tex]\( c = -9 \)[/tex]
Thus, this is not a perfect square trinomial.
To summarize, based on our calculations, the perfect square trinomials among the given polynomials are:
- [tex]\( 9a^2 - 30a + 25 \)[/tex]
All other polynomials are not perfect square trinomials.
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