yields the same remainder when divided by \( (x-2) \) and \( (x+3) \). What value o will satisfy the condition? Goal Your task is to apply the concept of remainder theorem in solving real life si ation. Role You will act as a donor of face mask in Aurora Audience School heads a teachers Situation You are the donor of face mask to the public schools in Aurora, and the numb of face masks you are giving to Maria Aurora National High School for teachers is expressed by a polynomial \( P(x)=p x 3+(p+3) x 2-9 x+30 \). It yie the same remainder when divided by \( (x-2) \) and \( (x+3) \). What value of \( p \) satisfy the condition? Product/ Performance and Purpose Answer the following questions: 1. Find \( p \) applying the concept of remainder theorem. 2. Using the value of \( p \) from question number 1, simplify \( P(x)=p x 3+(p+3) \) - \( 9 x+30 \). 3. How many face masks will be given to Maria Aurora National High Scho 4. Check your answer using long division and synthetic division methods

To solve the problem, we will apply the Remainder Theorem, which states that the remainder of a polynomial \( P(x) \) when divided by \( (x - a) \) is equal to \( P(a) \). ### Step 1: Find \( p \) applying the concept of the Remainder Theorem. Given the polynomial: \[ P(x) = p x^3 + (p + 3)x^2 - 9x + 30 \] We need to find \( p \) such that \( P(2) = P(-3) \). **Calculate \( P(2) \):** \[ P(2) = p(2)^3 + (p + 3)(2)^2 - 9(2) + 30 \] \[ = p(8) + (p + 3)(4) - 18 + 30 \] \[ = 8p + 4p + 12 - 18 + 30 \] \[ = 12p + 24 \] **Calculate \( P(-3) \):** \[ P(-3) = p(-3)^3 + (p + 3)(-3)^2 - 9(-3) + 30 \] \[ = p(-27) + (p + 3)(9) + 27 + 30 \] \[ = -27p + 9p + 27 + 27 + 30 \] \[ = -18p + 84 \] **Set \( P(2) = P(-3) \):** \[ 12p + 24 = -18p + 84 \] Combine like terms: \[ 12p + 18p = 84 - 24 \] \[ 30p = 60 \] \[ p = 2 \] ### Step 2: Using the value of \( p \) from question number 1, simplify \( P(x) \). Substituting \( p = 2 \) into \( P(x) \): \[ P(x) = 2x^3 + (2 + 3)x^2 - 9x + 30 \] \[ = 2x^3 + 5x^2 - 9x + 30 \] ### Step 3: How many face masks will be given to Maria Aurora National High School? To find the total number of face masks, we can evaluate \( P(x) \) at a specific value of \( x \). Since the problem does not specify a value for \( x \), we can evaluate \( P(1) \) as an example: \[ P(1) = 2(1)^3 + 5(1)^2 - 9(1) + 30 \] \[ = 2 + 5 - 9 + 30 \] \[ = 28 \] Thus, if \( x = 1 \), Maria Aurora National High School will receive **28 face masks**. ### Step 4: Check your answer using long division and synthetic division methods. **Synthetic Division:** We can check the polynomial \( P(x) = 2x^3 + 5x^2 - 9x + 30 \) for \( x = 2 \) and \( x = -3 \). 1. **For \( x = 2 \):** - Coefficients: \( 2, 5, -9, 30 \) - Synthetic division with \( 2 \): \[ \begin{array}{r|rrrr} 2 & 2 & 5 & -9 & 30 \\ & & 4 & 18 & 18 \\ \hline & 2 & 9 & 9 & 48 \\ \end{array} \] Remainder = 48. 2. **For \( x = -3 \):** - Coefficients: \( 2, 5, -9, 30 \) - Synthetic division with \( -3 \): \[ \begin{array}{r|rrrr} -3 & 2 & 5 & -9 & 30 \\ & & -6 & 3 & 18 \\ \hline & 2 & -1 & -6 & 48 \\ \end{array} \] Remainder = 48. Both synthetic divisions yield the same remainder of 48, confirming that \( P(2) = P(-3) \). Thus, the value of \( p \