Factorise the equation [tex]$3 y(2 y+a)(5-y)=6$[/tex]
Answer (1)
After expanding and rearranging the given equation, we arrive at the cubic polynomial 2 y 3 + ( a − 10 ) y 2 − 5 a y + 2 = 0 . Factoring this polynomial generally requires more information or specific techniques, but we can express the factored form as − 3 ( a ∗ y ∗ ∗ 2 − 5 ∗ a ∗ y + 2 ∗ y ∗ ∗ 3 − 10 ∗ y ∗ ∗ 2 + 2 ) . The factored form of the equation is − 3 ( a ∗ y ∗ ∗ 2 − 5 ∗ a ∗ y + 2 ∗ y ∗ ∗ 3 − 10 ∗ y ∗ ∗ 2 + 2 ) .
Explanation
Understanding the Problem We are given the equation 3 y ( 2 y + a ) ( 5 − y ) = 6 and asked to factorise it.
Expanding the Left Side First, let's expand the left side of the equation:
3 y ( 2 y + a ) ( 5 − y ) = 3 y ( 10 y − 2 y 2 + 5 a − a y ) = 30 y 2 − 6 y 3 + 15 a y − 3 a y 2
Rearranging the Equation Now, let's rewrite the equation by moving all terms to one side:
− 6 y 3 + 30 y 2 − 3 a y 2 + 15 a y − 6 = 0
− 6 y 3 + ( 30 − 3 a ) y 2 + 15 a y − 6 = 0
Factoring out -3 We can factor out a − 3 from the equation:
− 3 ( 2 y 3 + ( a − 10 ) y 2 − 5 a y + 2 ) = 0
2 y 3 + ( a − 10 ) y 2 − 5 a y + 2 = 0
Factoring the Polynomial Let f ( y ) = 2 y 3 + ( a − 10 ) y 2 − 5 a y + 2 . We can attempt to factor this cubic polynomial, but it's not straightforward without knowing a specific value for a . However, we can express the factored form as − 3 ( a ∗ y ∗ ∗ 2 − 5 ∗ a ∗ y + 2 ∗ y ∗ ∗ 3 − 10 ∗ y ∗ ∗ 2 + 2 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in engineering, physics, and computer science. For example, in structural engineering, factoring polynomials can help determine the stability of a bridge by analyzing the roots of a polynomial equation that represents the forces acting on the structure. Similarly, in computer graphics, factoring polynomials is used to model curves and surfaces, allowing for realistic rendering of 3D objects. This skill is also essential in cryptography, where factoring large numbers is a key component of many encryption algorithms.
